In a Kitchen-aid fitted with a blade mix the butter, flour, eggs and salt, combine well.
Activate the yeast in the warm milk. Add to the dough and mix.
Butter and line a medium size oven proof baking dish. Spread half of the dough in it with the help of the blade of a knife. It's a very sticky dough so be patient and makes sure to reserve half of the dough to cover the top.
Sprinkle with a layer of the fontina cheese cubed and lay the prosciutto slices on top. Cover with the remaining dough. Spread it even. Beat the remaining egg with a fork to make a wash, you can add teaspoon of water. With a brush paint the whole brioche evenly.
Leave the brioche to raise in a warm place for 3 hours. Cover loosely with a canvas.
Bake in a preheated oven at 350F for about 40 minutes. Slice and eat warm.