In a small skillet, melt 2 tablespoons of the butter over medium heat, then add the mushrooms, and raise the heat a bit. Cook the mushrooms until tender and dark, about 8 to 10 minutes. Add the sage midway through the cooking time. Season the mushrooms with salt and pepper, to taste, after they're dark and tender, as salt slows the browning by drawing out liquids.
Meanwhile, in a second small skillet over medium heat melt another tablespoon of the butter. Add the onions, season with salt and pepper, to taste, and cook until tender, 10 to 12 minutes. Stir in the sauerkraut, cook until warmed through, then turn off the heat. While the onions cook, heat a cast iron pan or griddle over medium-high heat. Season the chicken liberally with Montreal Steak Seasoning. Add the extra-virgin olive oil to the pan or griddle and cook the chicken, turning once, until cooked through, about 12 minutes. Remove the chicken to a plate. Wipe down the skillet or griddle, and reduce the heat to medium-low. Thinly slice the chicken on an angle.
Add the sour cream, relish, and ketchup to a small bowl and stir in the fresh herbs.
Lightly butter 1 side of the bread slices with the remaining tablespoon of butter. Build the sandwiches with buttered sides facing out, in this order: dressing, cheese, mushrooms, sliced chicken, sauerkraut with onions, cheese, and ending with the dressing to set in place. Grill the sandwiches for about 3 minutes on each side. Serve with pickles and salted radishes.