Preheat oven to 425°F. Combine parsnips, 2 tsp vegetable oil, and a pinch of salt and pepper on baking sheet and toss. Roast until parsnips are tender and golden brown, 18 to 20 minutes.

Heat 1 tbsp vegetable oil in large nonstick skillet over medium-high heat. Add mushrooms and cook, undisturbed, until starting to brown, 2 to 3 minutes. Add garlic and continue to cook until fragrant, 1 to 2 minutes. Transfer to plate and wipe skillet clean.

Stir together just half the mayo, Dijon, horseradish, and a pinch of salt and pepper in small bowl. Set aside horseradish mayo for next step.

Combine remaining mayo and caraway seeds in separate small bowl and set aside. Place sourdough on clean work surface and spread caraway mayo on each slice. Flip sourdough over and spread just 2 tsp horseradish mayo on other side of each slice. Divide cheese, cooked mushrooms, and sauerkraut between 2 sourdough slices. Close sandwiches with remaining slices, caraway side out.

Heat 1 tbsp vegetable oil in same skillet over low heat. Add sandwiches and cook until cheese is melted and bread is golden brown, 5 to 6 minutes per side. (TIP: If your bread is getting too dark but your cheese hasn’t melted yet, lower your heat.)

Cut mushroom reubens in half diagonally and place on plates. Serve with parsnip fries and any extra horseradish mayo. Tuck in!

For 4-serving meal, double all ingredients. You may need to cook in batches, in larger cooking vessels, or for longer cook times. Follow visual cues in 2-serving recipe.