Step 1

Make the dressing: In a medium bowl, using an immersion blender, process the mayonnaise, ketchup, horseradish, paprika, pickles and garlic, pulsing until well combined. (You can also use a food processor, but consider doubling the batch for easier processing.) The yield is about 1/2 cup; you'll have extra for more sandwiches or other uses. Transfer to an airtight container and refrigerate until needed.

Step 2

Make the mushrooms: In a small bowl, stir together the mustard, coriander, mustard seeds, salt, red pepper flakes, allspice and a few grinds of black pepper until thoroughly combined.

Step 3

Coat the bottom of a 12-inch nonstick skillet with a thin layer of oil and set over medium heat for several minutes, or until you feel a decent blast of heat when holding your hand an inch or so above the skillet. Arrange the mushrooms in the skillet in two layers. Wrap the bottom of a cast-iron skillet or a plate (weighted with heavy cans) in foil and place on top of the mushrooms; the goal is to press the liquid out of the mushrooms. Cook, undisturbed, for 6 minutes. Remove the skillet or plate, flip the mushrooms over, arranging them so the thickest slices are toward the middle, or the hottest part of the pan. Return the skillet or plate on top of the mushrooms and cook until the mushrooms have significantly reduced in size and almost all the liquid has evaporated, another 5 to 6 minutes. If there's still liquid in the skillet, uncover the mushrooms, briefly increase the heat to medium-high and cook until the water is mostly gone.

Step 4

Transfer the mushrooms to a platter and spread an even, thin layer of the mustard mixture on one side of each mushroom slice. (On the slices from the top halves of the mushrooms, the mixture adheres best to the exterior rather than the flesh.)

Step 5

Wipe out the skillet, add another thin coating of oil and return to medium heat (no need to preheat more than 1 minute or so, as it will still be quite hot). Place the mushrooms in a single layer, coated side down, and cook until browned and crisp in spots, 2 to 3 minutes, working in batches. A little of the coating may come off; that's okay — just pile it on when assembling the sandwiches. Wipe the skillet clean.

Step 6

Make the sandwiches: Spread about 1 tablespoon of the dressing on each of the 4 slices of bread. On 2 of them, build the sandwiches, evenly dividing the cheese, mushrooms and sauerkraut, followed by the other 2 slices of bread.

Step 7

In the skillet over medium heat, melt the butter until bubbling slightly. Griddle the sandwiches on both sides (do one at a time if you're tight on space) until golden brown and crisp, 2 to 3 minutes per side. Cut in half and serve.