Grilling the corn, onion, and peppers for the calabacitas will give them great flavor. Alternatively, you can broil the vegetables in the oven on a rimmed baking sheet until they become lightly charred.
- 2-1/2 Tbs. olive oil; more for the grill
- 1 small yellow onion, sliced into 1/2-inch-thick rings, skin intact
- 1 mild green chile, such as Hatch or Anaheim
- 1 large ear corn, husked
- Kosher salt and freshly ground black pepper
- 2 small summer squash, diced (about 2-1/2 cups)
- 4 medium cloves garlic, thinly sliced
- 1 tsp. toasted cumin seeds, crushed
- 1/4 packed cup fresh cilantro leaves, coarsely chopped (about 1/2 oz.)
- 2-3/4 cups coarsely grated Monterey Jack
- 2-3/4 cups coarsely grated sharp yellow Cheddar
- 16 store-bought or homemade 6-inch flour or corn tortillas
- Salsa (optional)
- Prepare a medium-high (400°F to 450°F) gas- or charcoal-grill fire.
- Oil the grill. Lightly brush the onion, chile, and corn with 1 Tbs. of the oil, and sprinkle generously with salt and pepper. Grill the vegetables until they become charred, turning them occasionally and removing them from the grill as they become done, 8 to 9 minutes for the onions, 11 to 12 minutes for the chile, and 15 to 16 minutes for the corn. Set aside until cool enough to handle. Remove the skin from the onion; coarsely chop, peel and seed the chile; and cut the corn from the cobs.
- In a large cast-iron skillet, heat 1 Tbs. of the oil over medium heat. Add the squash, 1 tsp. salt, and 1/4 tsp. pepper, and cook, stirring occasionally, until crisp-tender and golden in places, about 6 minutes. Add the garlic and the grilled onion, chile, and corn, and continue to cook, stirring until the garlic is fragrant, about 2 minutes. Add the cumin, and toss to combine. Remove from the heat, and transfer to a medium bowl. Fold in the cilantro, season to taste with salt and pepper, and set aside. Wipe the skillet clean.
- Position a rack in the center of the oven, and heat the oven to 200°F. Line a large rimmed baking sheet with parchment, and put in the oven.
- Combine the cheeses in a medium bowl. Heat the remaining 1/2 Tbs. oil in the skillet over medium heat. Put a tortilla in the skillet, and working quickly, apply even layers of 1/2 cup cheese, 1/3 cup squash, and 2 Tbs. cheese. Top with another tortilla. Cook, turning several times as necessary, until the cheese is melted and the tortillas are lightly charred, 3 to 4 minutes. Repeat with the remaining tortillas, cheese, and squash mixture. As they become done, transfer the quesadillas to the baking sheet in the oven to keep warm.
- Slice the quesadillas in quarters, and serve with your favorite salsa, if you like.