Make the filling:
In a medium bowl, combine the cream cheese, sugar, and cinnamon and beat with a hand mixer on high speed until smooth, scraping the sides of the bowl as necessary, about 1 minute. Gently stir in the sour cream and apple until just combined.
Make the French toast:
Create a pocket in each slice of bread by inserting a sharp knife into the center of the top crust and then working the knife in both directions, cutting to within 3/4 inch of the sides and bottom (be careful not to puncture the sides).
Squeeze the bread gently to part the opening. Spoon in 2 or 3 Tbs. of the prepared filling until the pocket is full but not bursting; the amount will depend on the size of the bread. Tap the bread on the counter to settle the filling. Wipe any extra filling from the opening with a clean paper towel.
In a medium bowl, whisk the eggs and cream. Dip each slice of stuffed bread into the egg mixture, soaking each side for about 1 min. to coat well and evenly. Stand the pieces upright in a baking dish and drizzle with any remaining egg mixture. Cover and refrigerate.
When ready to serve, heat the oven to 375°F. Lightly grease a baking sheet. In a large skillet, heat the butter and oil over medium-high heat until the butter is melted and foamy. Sauté as many pieces of French toast as will fit comfortably in the pan at one time, turning once, until golden brown on both sides, about 2 minutes. per side; the bread will puff up a bit. Continue with all pieces. Put the sautéed French toast on a greased baking sheet and bake until the filling is heated through, 6 to 8 minutes.
Make the syrup:
In a medium saucepan, whisk the apple cider, brown sugar, cloves, and cinnamon. Bring the mixture just to a boil and then lower the heat and simmer until thickened and reduced to about 1 cup, 8 to 10 minutes. Stir in the butter until melted. Serve warm over stuffed French toast.