Preheat oven to 375°F with a rack in the center.

Cut the bacon into bite-sized pieces. (I like to use kitchen shears for easy cleanup.) Place into a cold 10-inch skillet that's oven-safe.

Set skillet over medium heat and cook, stirring occasionally, until fat is rendered and bacon is lightly crisped.

While bacon cooks, shred the cheddar on the large holes of a box grater. Crack the eggs into a medium mixing bowl. Add cream, salt, pepper, and nutmeg. Whisk until homogenous.

Transfer bacon to paper towels to drain.

Clean the skillet, reserving two tablespoons bacon fat if you would like.

Set cleaned skillet back over medium heat. Add butter or reserved bacon fat and minced shallot.

Cook, stirring frequently, until shallot is tender.

Pour egg mixture into skillet. Sprinkle evenly with bacon and cheddar.

By this point the frittata will have begun to set underneath. Transfer from stovetop to oven.

Bake until puffed and just set throughout, about 12 minutes.

Let cool slightly in pan, then use a spatula to ensure frittata releases from pan and slide onto a cutting board.

Cut into four pieces.

Split each roll in half. Toast lightly if you like.

Spread each half with about ¾ tablespoon of butter.

Drape prosciutto evenly over buttered rolls.

Place a slice of frittata onto each bottom half.

Top with greens.

Close each sandwich and cut in half.