Caramelize the onions. Add sliced onion and 1/4 cup water to a large skillet over high heat. Cover and cook until the water has mostly evaporated and onions start to sizzle. Uncover, reduce the heat to medium-high, and continue to cook the onions while stirring frequently until they are soft and browned, about 20 minutes. Stir in sage and thyme in last 5 minutes of cooking. (Note: add more water a tablespoon at a time throughout this process, if desired, to keep the onions from sticking.)
Assemble the sandwiches. Brush melted butter on one side of each slice of bread. Lay bread, buttered side down, on a large plate or cutting board. On two of the slices, layer about 3/4 of the shredded cheese.
Next, layer caramelized onions, turkey, and apple slices on top of the cheese. Top with remaining cheese.
Make it melty. Heat a large non-stick skillet or griddle over medium heat. While the skillet heats, wrap the bottom of a large cast iron skillet in foil. (Note: If you have cast iron grill presses, you can skip wrapping a cast iron skillet and use those instead. You just need something weighted to help press and heat the sandwiches through.)
Add sandwiches to the heated skillet and gently lay the foiled cast iron skillet on top. Cook the sandwiches until golden brown on each side and heated through, about 3-4 minutes per side.
Slice sandwiches in half before serving and serve with butternut squash soup, if desired.
Notes
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