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Preheat oven to 400 degrees.
Peel the Kohlrabi until you have removed the fibrous outer layer; then coarsely shred it, using a box grater. Remove core from pepper and slice into paper thin strips.
In a large bowl, whisk the mayonnaise with 2 tablespoons olive oil, vinegar, mustard, sugar, 1 1/2 teaspoons sea salt and 3/4 teaspoon pepper. Add kohlrabi and pepper and toss well. Cover and refrigerate.
Clean mushrooms with a damp towel. Using two forks, roughly shred the stems and caps into pieces. Set on a parchment paper-lined baking sheet.
Drizzle with 1 tablespoon oil, and sprinkle with paprika, salt, cayenne and garlic. Toss evenly to coat the mushrooms; then bake for 20 minutes, or until mushrooms are a bit crispy and brown on the edges.
Heat remaining 1 tablespoon oil in a large saute pan over medium high. Transfer cooked mushrooms to pan and add BBQ sauce. Stir and cook for 3 to 5 minutes, until mixture is thick and fragrant.
Cut buns in half and lightly toast. Add a spoonful of mushrooms to each buns and top with kohlrabi slaw.