Warm the water to between 105 and 110 deg F. Add 1/2 tsp sugar and the active dry yeast. Stir briefly and then set aside for 5 minutes to activate.
In the bowl of a stand mixer combine the flour, 1/2 cup sugar, and salt. Stir together and attach the dough hook. Add the beaten eggs along with the activated yeast.
Mix, using the dough hook, until the dough starts to come together. Scrape down the sides occasionally to encourage the dough to come together. If the dough is too sticky and sticks to the side without forming a ball, then add another tablespoon or two of flour.
Once a ball has formed, continue mixing with a dough hook for about 7 minutes. After kneading, the dough will be sticky and stick to the sides of the mixer.
After the dough has kneaded for about 7 minutes begin to add the butter. This process will take 10 minutes or more. With the mixer running on low add 2 tablespoons of the softened butter. Scrape down the sides of the bowl with a spatula occasionally to encourage the butter to incorporate.
When the butter has mostly been incorporated add another 2 tablespoons and repeat the process. Continue this process until all of the butter has been added.
Once all of the butter has been added to the dough add another tablespoon or two of flour. Use a spatula to scrape the dough off the sides of the mixer bowl and come together in a ball.
Generously flour the counter. Turn the dough onto the floured counter. The dough will be very sticky and soft. Sprinkle with a generous amount of flour and begin to knead the dough. Use as much flour as you need to prevent the dough from sticking to the counter.
Knead the dough until it comes together into a soft dough adding flour as needed. Transfer the dough to a greased bowl. Drizzle a little more oil on top of the dough and rub it all over the dough. Cover with cling film or a tea towel and then set in a warm place to rise for 1 hour.
After the brioche has risen for 1 hour turn onto a lightly floured counter. Knead the dough back into a smooth ball to remove the air bubbles. Transfer the dough back to the greased bowl. Drizzle a little more oil on top of the dough and rub it all over the dough. Cover tightly with cling film and refrigerate overnight.
Peach Filling
Finely dice the peaches. In a saucepan melt the butter over medium heat and then add the peaches. Add the peach nectar bring the mixture to a boil. Turn down the heat and cover with a lid. Let the peaches cook until very soft, about 5 to 8 minutes.
Meanwhile, combine the sugar, cornstarch, cinnamon, allspice, and salt in a small bowl. Whisk together until there are no lumps. Set aside. Add the plain gelatin to a small bowl or cup. Add the water and stir together. Set aside to bloom.
When the peaches are softened add the sugar mixture and stir to combine. Bring the peach filling back up to a boil. Add the bloomed gelatin and stir until it has completely melted. Transfer the peach filling to a bowl. Cover with cling film placed directly on top of the filling. Refrigerate overnight until set.
Crisp Topping
Preheat the oven to 350 deg F. Line a baking sheet with parchment paper. In a mixing bowl combine the flour, oats, brown sugar, cinnamon, allspice, nutmeg, and salt. Stir together. Add the melted butter and stir to combine.
Use your hands to crumble the oat mixture onto the baking sheet. Spread the crumbs out evenly on the baking sheet. Break up any large clumps. Bake for 10 minutes. Use a spatula to flip the crumbs so the brown evenly. Bake for another 5 to 10 minutes until well browned. Let cool before using.
Icing
To make the icing combine the icing sugar, butter, and peach nectar in a bowl. Mix together thoroughly. Cover the icing with cling film to prevent it drying out.
Fry and Assemble the Doughnuts
Lightly flour the counter and place the cold refrigerated dough on top. Dust lightly with flour. Use the rolling pin to press down on the dough and roll it out into a large oval shape 1/4 inch thick.
Use a 2-inch cutter to cut as many doughnuts out on the first roll. You should be able to get 11 to 12 doughnuts on the first roll. Gather up the scraps and press into a ball. Roll the dough out and cut out 2 to 3 more donuts.
Transfer the donuts to a parchment lined baking sheet. Cover with a tea towel and let rise until doubled in size. As the dough is cold this can take 60 to 80 minutes.
In a deep wide skillet or saucepan add 2 inches of vegetable oil. Bring the oil temperature up to between 340°F and 350° F. Maintain this temperature while frying the donuts. Line a baking sheet with a rack or paper towels to drain.
Add 3 or 4 donuts to the hot oil. Let fry for 3 to 4 minutes until the bottom is browned. If large air bubbles form on top, then pop them before flipping. Flip over and continue to fry for another 3 to 4 minutes until the other side is browned. If desired, check doneness with an instant read thermometer. The temp should be over 180° F inside.
Use a slotted spoon to lift the doughnuts out of the oil and transfer to the rack-lined baking sheet. Repeat with the remaining doughnuts. Let cool before filling.
Fit a pastry bag with a large tip. Transfer the peach filling to the pastry bag. Cut a slit on the side of each donut. Use your finger or the handle of a wooden spoon to widen the opening and form a pocket inside the doughnut.
Pipe some of the peach filling into each doughnut. Spread a thin layer of icing on top and sprinkle generously with the crisp filling. Press the crisp filling onto the icing so that it will adhere. Serve and enjoy!