Roasted Winter Veggie Burgers

Name: Roasted Winter Veggie Burgers
Ingredients: 2 teaspoons extra-virgin olive oil; 1 large sweet potato unpeeled, scrubbed, and diced; 1 medium red onion, half minced and half thinly sliced; 2 teaspoons apple cider vinegar; 1 teaspoon sea salt, or to taste; 1 medium parsnip, grated; 1 medium carrot, grated; 1 large garlic clove, minced; 1 1/2 cups store-bought or homemade hummus; 6 tablespoons store-bought or homemade sun-dried tomato pesto; 2 teaspoons finely chopped fresh rosemary; 1 teaspoon freshly ground black pepper, or to taste; 1/4 teaspoon ground cayenne, or to taste; 1 (14-ounce) package firm tofu, well drained, gently squeezed dry, and crumbled; 1 cup plain dry whole-wheat breadcrumbs; 8 slices Cheddar cheese; 8 whole-grain hamburger buns, split, or 8 whole-grain sandwich rounds (thins), toasted; 4 slices large beefsteak tomato (optional); Baby arugula, for serving