Heat the vegetable oil in a large nonstick skillet over medium-high heat. Add the walnuts, shallot and garlic and cook, stirring, until softened, about 1 minute. Add the mushrooms and season with salt and pepper. Cook, stirring occasionally, until lightly browned and tender, about 5 minutes. Add the Worcestershire sauce and paprika and cook, tossing to coat, until reduced, about 10 seconds. Remove from the heat and stir in the mustard. Transfer the mixture to a large bowl.
Meanwhile, microwave the sweet potato until tender, 8 to 10 minutes. Let cool, then peel and transfer to a large bowl. Add the quinoa and beat with a mixer on medium-high speed until combined. Beat in the breadcrumbs, 1/2 teaspoon salt and a few grinds of pepper. Form into four 4-inch-wide patties and freeze until firm, 20 minutes.
Preheat a grill to medium high. Oil the grates. Grill the patties until marked, about 3 minutes per side. Grill the buns. Serve the burgers on the buns with onion and sprouts.