Slice peaches over a heavy-duty saucepan to catch juices; discard pits. Stir in sugar, cinnamon, and nutmeg; cook over medium heat until bubbly and sauce reaches a syrupy consistency, stirring occasionally, about 10 minutes. Remove from heat; set aside.

Meanwhile, cut off and discard ends of rolls. Cut rolls into 1 1/4-inch-thick slices. Cut a horizontal pocket into each slice, leaving three sides intact; set aside.

Stir mascarpone, confectioners' sugar, and lemon zest together until smooth; transfer to a small resealable plastic bag. Snip off one corner of the bag; pipe as much filling into pocket of each bread slice as fits without overflowing.

Whisk eggs, milk, and vanilla together in a shallow bowl.

Melt butter with oil in a large nonstick skillet over medium heat. Dip stuffed bread slices in egg mixture. Add to the skillet; cook until browned on both sides.