Heat the oven to 450 F. In a large bowl combine the squash, 1/4 cup (60 ml) olive oil, smoked paprika and season with salt and toss well. Transfer the mixture to a parchment-lined baking sheet and evenly spread out, roast until tender and golden 15 – 20 minutes. Toss slightly to ensure even cooking. Remove from the oven and let cool.
While squash is roasting, heat 1/4 cup (60 ml) olive oil over medium-high heat, add the onion and season with a little salt. Cook, stirring frequently, until the onions are well softened and start to turn golden brown (about 15 minutes). Add the vinegar and brown sugar, stir and reduce until consistency looks like honey (about 15 minutes).
Combine roasted squash and onions in a bowl and smash with a fork until combined. Taste for seasoning, at this stage you can add the pepper. If it feels too acidic, feel free to add a little honey or maple syrup.
Assemble sandwiches by placing roasted squash mixture on one slice of bread. Spread 1/4 of the cheese on the other slice. Put both slices of bread together to form a sandwich. Brush outside with butter. Repeat to make 3 more sandwiches.
Heat grill pan over medium heat; place sandwich in the pan and top with another heavy pan to compress the sandwich. Reduce heat to medium; cook for 3 to 4 minutes. Flip sandwich over and cook until cheese is melted (3 to 4 minutes).
