Preheat oven to 350˚F.
Melt butter into EVOO in a small pot over medium heat. When it foams, add garlic and remove from heat.
Cut a round piece about 2 1/2 to 3 inches wide into each roll top but dont cut all the way through. Hollow out the insides of the roll; remove and reserve the cut-outs. Brush the insides of the rolls and the bottoms with a little of the garlic butter.
Arrange roll tops on a baking sheet and line each cavity with two slices of ham, folding it to fit in the cavity but only slightly overlap the edge. Mix the two cheeses together and distribute between the rolls. Crack an egg on top of each pile of cheese and season with a little pepper. Place the reserved roll cut-outs on the baking sheet with the filled rolls and bake for 15 minutes or to desired doneness. The little tops just need to be toasted so remove those before the filled rolls are ready to come out.
Top eggs with greens (if using) set tops in place.