Preheat oven to 375 degrees Fahrenheit. Set out two rimmed baking sheets.
Drizzle 1 tablespoon olive oil into a large saute pan over medium heat. When hot, add the onions. Saute, stirring occasionally, until tender and browned here and there, about 5 minutes. Remove onions from pan and return pan to the burner. Reduce heat to medium and immediately add the remaining 1 tablespoon olive oil and the squash.
Cook, stirring occasionally, until most of the squash pieces have been seared golden brown on at least one side and are beginning to soften, about 5 minutes. Reduce heat to medium-low and add water, brown sugar, cumin, smoked paprika, chili powder, salt, and pepper. Stir gently to distribute water and spices and continue to saute, stirring occasionally, until the water cooks off, about 3 minutes. Remove from heat.
Place a flour tortilla on your prep surface and add 1/6 of the toppings to the bottom half of the tortilla, in a single layer. Start with the squash, then scatter about 3 tablespoons of black beans, then add some onions and finally 3-4 slices of the pepper jack cheese.
Fold the tortilla over to form a half-moon shape and carefully transfer to a baking sheet. Repeat with remaining tortillas. Make sure the quesadillas sit in a single layer on the baking sheets - I fit three to a sheet. Brush the quesadillas on both sides with the 1 tablespoon olive oil; this helps them bake up a little more crisp and golden.
Bake until the cheese has melted and the tortilla is golden brown and crisp in spots. Using a pizza cutter, cut each quesadilla into thirds. Transfer to plates and serve with assorted toppings, if desired.