Lay 4 slices of the bread on a flat work surface. Top each slice with at least ¼ cup of the Apple Pie Compote, spreading it evenly. Top with the remaining slices of bread to form 4 sandwiches.
Place the cereal crumbs on a flat dinner plate. In a large bowl, beat together the eggs, half-and-half, vanilla, and sugar until very smooth. Dip each sandwich into the egg mixture, turning it over to make sure it is well-soaked and completely coated. Dredge both sides of each sandwich completely in the cereal crumbs.
Heat 2 tablespoons of the butter and 2 tablespoons of the oil in a large 10- or 12-inch nonstick skillet or frying pan, until the butter melts and starts to sizzle.
Cook, 2 sandwiches at a time, until they are slightly puffed and crisp and golden brown all over, 2 to 3 minutes per side. Carefully wipe out the pan with a paper towel, and heat together the remaining 2 tablespoons butter and 2 tablespoons oil. Cook the remaining 2 sandwiches until crisp and golden brown.
Remove the sandwiches from the pan and place on a cutting board. Cut each sandwich in half on the diagonal to expose the apple filling. Stack two halves onto an individual serving plate and sprinkle liberally with confectioners' sugar. Repeat with the remaining sandwiches and serve immediately.