Remove silvery membrane from tenderloin with a sharp knife; reserve trimmings. Cut the small "head" pieces of tenderloin where there is a natural separation; reserve. (This is optional.) Chop reserved trimmings very fine; set aside.

Cut tenderloin in half crosswise to make 2 equal portions. Cut each half in half lengthwise to make 4 pieces. Transfer to a bowl; refrigerate until needed.

Melt butter in a pot over medium-high heat. Add reserved trimmings; cook and stir until caramelized and nicely browned, 4 to 5 minutes. Stir in broth and gelatin, then stir in water; simmer and stir over medium or medium-low heat until liquid is reduced by half, about 1 1/2 hours. Add more water if it reduces too quickly.

Meanwhile, place pork sections between two pieces of heavy plastic wrap on a solid, level surface. Firmly pound to a thickness of about 1/4- to 1/8-inch thick. Remove top sheet of plastic wrap; flip each piece pork. Season generously with salt and black pepper; dust very lightly with about 2 teaspoons flour. Flip each piece pork; sprinkle with black pepper. Press 3 sage leaves onto surface of each piece pork; cover completely with prosciutto, cutting or tearing strips to fit. A bit of overlap is okay.

Return sheet of plastic wrap over pork; pound very lightly to make sure prosciutto sticks to pork pieces. Transfer to a plate; cover with plastic wrap. Refrigerate until chilled, at least 30 minutes.

Strain pork broth into a bowl; discard solids.

Heat olive oil in a skillet over high heat until shimmering. Place pork pieces into the skillet, prosciutto-side down; cook about 3 minutes. Carefully flip pieces; cook 1 minute more. Remove skillet from the heat. Transfer pork to warm serving plates.

Blot excess oil from the skillet with paper towels. Add wine and pork broth; cook over high heat until thickens and reduces by about half, 4 to 5 minutes.

Spoon hot sauce over pork.