Washington State Burger (Aka Wild Mushroom-Cheddar Burger)

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2 tablespoons olive oil

1 tablespoon unsalted butter

12 ounces assorted mushrooms, such as cremini, lobster, chanterelles, and stemmed shiitakes, coarsely chopped

1 small shallot, finely diced

Kosher salt and freshly ground black pepper

1 tablespoon chopped fresh thyme leaves

2 tablespoons chopped fresh flat-leaf parsley

1 1/2 pounds ground chuck (80 percent lean) or ground turkey (90 percent lean)

2 tablespoons canola oil

4 slices sharp Cheddar

4 hamburger buns, split; toasted if desired