Stir warm milk, yeast and sugar in a large bowl. Add 4 cups flour, salt, egg and butter and mix into a shaggy dough; knead for 8 to 10 minutes until smooth and elastic, using the extra flour only as needed to prevent sticking.
Cover and let rise in a warm spot for 60 to 75 minutes until doubled. Line two rimmed sheet pans with parchment and divide the dough in half.
Roll each half into a 12-by-10-inch rectangle. Spread each with 1 tablespoon Dijon, leaving a 1/2-inch strip bare along one long edge, then scatter half the ham and cheese evenly over each.
Roll each rectangle from the opposite long edge into a 12-inch log and pinch the seam closed. Cut each log into 12 equal 1-inch slices and place cut side up on the pans, spaced about 1 inch apart.
Cover loosely and let rise for 25 to 35 minutes until puffy. Heat oven to 350°F during this second rise.
Bake for 25 to 30 minutes, switching pan positions halfway through, until golden, the bread centers reach about 190°F and the ham filling reaches at least 165°F. Cool on the pans for 10 minutes before serving.
Serve 24 filled party rolls, two per portion. Refrigerate leftovers promptly in a covered container and reheat covered at 325°F until the filling reaches 165°F.