In large nonstick skillet over medium heat, melt 1 tablespoon butter. In wide, shallow bowl, whisk together 1 egg, half and half, salt, cinnamon, and 1 tablespoon maple syrup to make a French toast style batter. Dip 1 side of each bread slice into batter, being careful to keep other side of bread dry. Place bread, French-toast-side-down, in buttered skillet and grill this side until golden. Meanwhile, in small nonstick skillet, fry remaining egg over easy with a bit of butter so yolk is slightly runny. Remove bread and egg from skillets; keep warm.
Spread half of cream cheese on each “French toast” side of bread slices. Melt remaining butter over medium-low heat in large skillet. Place 1 bread slice in the skillet, so French toast/cream cheese side is facing up.
Sprinkle 1/4 cup cheddar cheese over. Mix butternut squash puree with remaining tablespoon maple syrup and spread half of it over cheese. Sprinkle with some of the thyme and top with the egg, bacon, remaining squash puree, additional thyme, and remaining cheddar. Cover with other bread slice, so French toast/cream cheese side is down. Grill sandwiches on each side over low heat, until golden brown and gooey.
Recipe adapted from www.grilledcheeseacademy.com