Melt the butter in a saucepan over low heat. Slowly cook the butter on low for several minutes until the milk solids separate from the butter fat. The milk solids will settle on the bottom of the pot and a layer of crusty foam will form on top of the clear melted butter. Use a skimmer or spoon to scoop off the crusty foam. Carefully pour the clear melted butter into a separate jar being careful not to disturb the milk solids on the bottom of the pan. Set aside.
Line one large ramekin with two layers of cling film if you want to be able to remove the potted ham. Alternatively, you can use 2 or 3 smaller ramekins as needed. Finely shred or mince the ham. Use forks to shred the ham or a food processor to mince it if you prefer. In a mixing bowl, combine the ham with the finely diced pickles, chopped fresh dill, kosher salt, and pepper. Stir together.
Add about three quarters of the clarified melted butter to the ham and stir to combine. Reserve the remaining butter. Spoon the ham mixture into the prepared ramekin, pressing down to pack the ham firmly into the ramekin. Smooth the top layer of the ham. Pour the remaining butter on top of the ham to seal it in. Chill for several hours or until firm.
To serve, remove the potted ham from the ramekin. Allow the ham to soften at room temperature for up to an hour before serving if desired. Serve with rye bread and pickles. Enjoy!