Heat the olive oil and melt the butter in a large saute pan over high heat until almost smoking. Add the mushrooms and cook, stirring occasionally, until soft, about 5 minutes. Add the shallot, season with salt and pepper, and cook until the mushrooms are golden brown, about 5 minutes. Stir in the thyme and parsley and transfer to a bowl.
Divide the meat into 4 equal portions (about 6 ounces each). Form each portion loosely into a 3/4-inch-thick burger and make a deep depression in the center with your thumb. Season both sides of each burger with salt and pepper.
Heat a gas grill to high. Brush the burgers with canola oil and grill until desired doneness is reached. Top each with a slice of cheese and cover during the last minute of cooking.
Place the burgers on the bun bottoms and top each burger with a large spoonful of the mushrooms. Cover with the bun tops and serve immediately.
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