For the roasted winter veggie burgers: Heat a large nonstick skillet or cast iron over medium heat. Add the olive oil, sweet potato, minced onion, vinegar, and 1/4 teaspoon of the salt and saute until the sweet potato is nearly tender, about 15 minutes. Add the parsnip and carrot, cover, and cook, stirring occasionally, until all vegetables are tender, about 6 minutes. Stir in the garlic and saute for 1 minute. Set aside, uncovered, to cool slightly.
Preheat the oven to 450 degrees F.
Combine the hummus with the sundried tomato pesto. Set aside 1/2 cup. Add 1 cup of the hummus, the rosemary, black pepper, cayenne, and the remaining 3/4 teaspoon salt to a medium bowl and stir until combined. Stir in the tofu and breadcrumbs until combined. Stir in the sweet potato mixture and thoroughly combine with your hands.
Firmly form the mixture into 8 (4-inch-wide) patties, about 2/3 rounded cup mixture each.
Place the patties on a parchment paper-lined baking sheet. Roast until well browned on top, about 10 minutes, then flip and cook for 8 minutes more until they are cooked through and slightly crisp. Add a slice of cheese to each and return to oven until melted, about 2 minutes.
Place the patties into the buns with the arugula, tomato (if using), sliced onion, and the remaining 1/2 cup hummus, and serve.