For the grilled mushrooms and onions: Prepare a charcoal grill for direct and indirect heat. Brush the portobello mushrooms, shiitake mushrooms and onions with oil on both sides and season with salt and pepper. Grill over direct heat on both sides until slightly charred and just cooked through, 4 to 5 minutes per side. Reserve for the burgers.
For the burgers: Remove the planks from the water and place them over direct heat for about 2 minutes per side. Remove the planks from the grill and brush the top side with oil.
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. Brush with canola oil and sprinkle with salt and pepper. Put the burgers on the grill over direct heat and cook until nicely charred, about 3 minutes. Flip and cook until the second side is charred, about 3 minutes. Place 2 burgers on each cedar plank. Put the planks over indirect heat, brush each burger with 1 tablespoon Red Wine Barbeque Sauce and top each with 2 slices of cheese. Cover the grill to melt the cheese and finish cooking, about 5 minutes.
Put the burgers on the bottom buns and top each with more barbeque sauce, some mushroom caps, some grilled onions and the top bun. Serve immediately.
Combine the ketchup, 1 cup water, ancho chile powder, honey, molasses, sugar, mustard, paprika, vinegar, Worcestershire sauce, chile de arbol, and chipotle chile in a medium saucepan over medium heat. Bring to a simmer and cook until thickened, about 30 minutes, stirring occasionally.
Meanwhile, bring the wine to a simmer in a small saucepan over medium heat. Simmer until reduced to 1 cup, about 20 minutes. Stir the reduced wine into the sauce. Season with salt and pepper. Pour into a bowl and allow to cool at room temperature. (The sauce will keep for 1 week in the refrigerator stored in a tightly sealed container.)