Make the dough. Combine milk with yeast and sugar in a medium bowl. Let sit for 5 minutes or until yeast is activated and mixture is foamy.
In the bowl of a stand mixer, combine yeast mixture with flour, salt, and softened butter. Mix until a dough forms. Continue to knead for 5-7 minutes or until smooth. Alternatively, knead by hand. Transfer dough to a clean bowl and wrap with plastic wrap. Let sit in warm area for 1 hour or until doubled in size.
Preheat oven to 400 degrees F.
Line a baking sheet with parchment paper. Lay prosciutto on the baking sheet and bake for 7-8 minutes or until crispy. Let cool then crumble into small pieces.
Melt 1 tablespoon butter in a large saute pan over high heat. Add eggs and cook until halfway cooked. Remove from heat and stir in ricotta. Season with salt and pepper and let cool.
Add chopped chives and crispy prosciutto to eggs, stirring until combined. Set aside.
Reduce oven temperature to 375 degrees F. Grease a 9x13-inch baking pan with butter.
Lightly dust a work counter with flour. Punch down dough and roll into a 12x18-inch rectangle. Sprinkle 1 cup cheddar evenly over dough. Spread egg mixture on top and sprinkle remaining cheese. Roll dough with the long side facing you so that you end up with an 18-inch log. Cut log into 12 rolls that are 1 ½ inches wide.
Place rolls cut side facing up in prepared baking pan. Loosely cover and let sit in a warm place for 30 minutes.
Bake rolls for 25-30 minutes or until golden brown. Remove from oven and let cool slightly. Serve warm.