1 2 1/2- to 3-pound kabocha or other yellow-fleshed squash, peeled, seeded and cut into pieces 1/8- to 1/4-inch thick
3/4 cup extra-virgin olive oil
1/2 teaspoon dried chile flakes, more to taste
3 teaspoons kosher salt
1 yellow onion, peeled and thinly sliced
1/4 cup apple cider vinegar
1/4 cup maple syrup
4 slices country bread, 1-inch thick
1/2 cup ricotta, goat cheese, feta or mascarpone
4 tablespoons chopped mint
Heat the oven to 450. Combine the squash, 1/4 cup olive oil, chile flakes and 2 teaspoons of salt in a bowl and toss well. Transfer the mixture to a parchment-lined baking sheet and cook, stirring every few minutes, until tender and slightly colored, about 15 minutes or a little longer. Remove from the oven.
Meanwhile, heat another 1/4 cup olive oil over medium-high heat, add the onions and remaining teaspoon salt and cook, stirring frequently, until the onions are well softened and darkening, at least 15 minutes. Add the vinegar and syrup, stir and reduce until syrupy and broken down, again at least 15 minutes or so; the mixture should be jammy.
Combine squash and onions in a bowl and smash with a fork until combined. Taste for seasoning.
Add the remaining oil to a skillet over medium-high heat. Working in batches if necessary, add bread and cook until just golden on both sides, less than 10 minutes total; drain on paper towels. Spread cheese on toasts, then top with the squash-onion mixture. Sprinkle with coarse salt and garnish with mint.