Submerge the cedar planks in cold water and soak for at least 1 hour.
Preheat the oven to 400°F.
Heat the olive oil in a large sauté pan over high heat. Add the mushrooms and cook, stirring occasionally, until they begin to soften, about 5 minutes. Add the shallots and cook until the mushrooms are golden brown, about 5 minutes longer. Stir in the thyme and season with salt and pepper.
Remove the planks from the water, put them on a baking sheet, and place them in the oven for 15 minutes. Remove and brush the tops of the planks with canola oil.
Raise the oven temperature to 425°F.
Divide the meat into 4 equal portions. Form each portion loosely into a 3/4-inch-thick burger and then make a deep depression in the center with your thumb. Season both sides of each burger with salt and pepper.
Place 2 burgers on each plank, leaving a few inches between the burgers. Brush the tops of the burgers with some of the barbecue sauce and place the planks in the oven. Cook, basting with some of the sauce every few minutes, until the burgers reach the desired doneness, 10 to 12 minutes for medium-rare. Top each burger with 2 slices of cheese during the last minute of cooking.
Remove from the oven and top each burger with a large spoonful of the mushrooms. Serve the warm buns on the side.
RED WINE BARBECUE SAUCE
Heat the oil in a heavy medium saucepan over medium heat. Add the shallots and garlic and cook until soft, 3 to 4 minutes. Add the ground ancho, paprika, and ground chile de árbol and cook for 30 seconds. Raise the heat to high, add the wine, and boil until completely evaporated.
Add the ketchup and 1/2 cup water and bring to a boil. Lower the heat and simmer for 5 minutes. Add the mustard, vinegar, Worcestershire, chipotle, brown sugar, honey, and molasses and simmer, stirring occasionally, until thickened, about 10 minutes.
Transfer the mixture to a food processor and puree until smooth. Season with salt and pepper. Pour into a bowl and allow to cool at room temperature. The sauce will keep for 1 week in a tightly sealed container stored in the refrigerator.