Heat oven to 325°F and set a wire rack over a sheet pan. Spread bread with mustard and assemble four two-slice sandwiches with ham, turkey and Gruyere divided evenly.
Press the sandwiches gently and cut each diagonally into two triangles. Whisk flour, baking powder and salt, then whisk in egg and milk to make a thick pourable batter.
Heat frying oil in a deep heavy pot to 340°F, using a pot large enough that oil fills it no more than halfway. Keep a thermometer in place and adjust the heat between batches.
Dip two sandwich triangles in batter, letting excess drip off, then lower carefully into the oil. Fry for 3 to 4 minutes, turning once, until golden; repeat in four total batches.
Transfer fried triangles to the prepared rack. Bake for 8 to 12 minutes until the filling at the center reaches 165°F, checking more than one piece.
Dust lightly with powdered sugar. Serve two triangles per person with 1 tablespoon blackberry jam alongside.