Heat the olive oil and 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.
Cook the burgers using the canola oil (see Cook's Note) and topping each one with a slice of cheese and a basting cover during the last minute of cooking.
Place the burgers on the bun bottoms and top each burger with chipotle ketchup, if using, and a large spoonful of the mushrooms. Cover with the bun tops and serve immediately.
I love ketchup as much as the next American does, but when I was designing the menu for Bobby's Burger Palace, I knew that I wanted to have a little something extra to offer in the squeeze bottles. The simple addition of pureed chipotle in adobo gives prepared ketchup an exciting smoky edge, while still keeping its sweet yet tangy identity intact.
Whisk together the ketchup, chipotle, salt, and pepper in a small bowl. Cover and refrigerate for at least 30 minutes to allow the flavors to meld. The sauce will keep for 1 week in a tightly sealed container in the refrigerator.
Toasted Burger Buns: I think the taste and the texture of buns are best when lightly toasted. To toast a bun on a grill, grill pan, or griddle, split the bun open, place it cut-side down on the grill, and grill until light golden brown, about 10 seconds. Alternatively, you can place the split bun halves cut-side up on a baking sheet and cook them under a preheated broiler until light golden brown, about 30 seconds. Keep a close eye on them and don't let them get too brown or else that texture will take on a life of its own and defeat the purpose of starting with a soft bun. IF USING A GRILL: Heat a gas grill to high or heat coals in a charcoal grill until they glow bright orange and ash over. Brush the burgers with the oil. Grill the burgers until golden brown and slightly charred on the first side, about 3 minutes for beef and 5 minutes for turkey. Flip over the burgers. Cook beef burgers until golden brown and slightly charred on the second side, 4 minutes for medium rare (3 minutes if topping with cheese; see step 3) or until cooked to desired degree of doneness. Cook turkey burgers until cooked throughout, about 5 minutes on the second side. IF USING A GRILL PAN: Heat a grill pan over high heat on top of the stove. Cook the burgers as for a grill, above. IF USING A SAUTE PAN OR GRIDDLE (PREFERABLY CAST IRON): Heat the oil in the pan or griddle over high heat until the oil begins to shimmer. Cook the burgers until golden brown and slightly charred on the first side, about 3 minutes for beef and 5 minutes for turkey. Flip over the burgers. Cook beef burgers until golden brown and slightly charred on the second side, 4 minutes for medium rare (3 minutes if topping with cheese) or until cooked to desired degree of doneness. Cook turkey burgers until cooked throughout, about 5 minutes on the second side.