Line a greased 8-in. square baking dish with pastry. Trim to 1/2 in. below edge of dish; flute edges. Line unpricked pastry shell with a double thickness of heavy-duty foil. Bake at 450° for 8 minutes. Remove foil; bake 5 minutes longer. Cool on a wire rack.
Meanwhile, in a large skillet, cook the onions, garlic, salt and pepper in oil over low heat for 25-30 minutes or until onions are golden brown, stirring frequently. Add cream and thyme. Bring to a gentle boil. Reduce heat; simmer for 2-4 minutes or until thickened. Cool slightly.
Pour into crust. Top with apple slices. In a small bowl, combine squash and egg; spread over top.
Cover and bake at 375° for 15 minutes. Uncover; arrange cheese in a lattice pattern over the top. Bake 15-20 minutes longer or until apple is tender. Let stand for 10 minutes before cutting.