Grilled Bison-Mushroom Burgers

Pulse mushrooms in a food processor, scraping down the sides once or twice, until finely chopped but not a paste.

Heat 1 tablespoon oil in a medium skillet over medium-high heat. Add the mushrooms and cook, stirring occasionally, until browned and the liquid has evaporated, about 5 minutes. Add 2 tablespoons vinegar, garlic powder and paprika; cook, stirring, for 2 minutes. Scrape into a medium bowl and spread across the bottom in a thin layer. Let cool for 15 minutes.

Meanwhile, preheat grill to medium-high.

Toss onion in a small bowl with the remaining 2 teaspoons each oil and vinegar and 1/4 teaspoon salt. Let stand, stirring occasionally.

Add bison (or beef) and the remaining 1/4 teaspoon salt to the mushrooms; mix well. Form into 4 equal patties, about 4 inches in diameter.

Grill the burgers, turning once, until an instant-read thermometer registers 155 degrees F, 8 to 10 minutes total. Grill the onions, turning once, until tender and lightly charred, 6 to 10 minutes total. Top the burgers with cheese and cook until it's melted, about 1 minute more. Serve the burgers on buns, topped with the onions.