Heat oven to 375°F. Line a rimmed baking sheet with foil and lightly oil it. Wipe mushrooms clean, twist out the stems, and finely chop the stems for the filling.
Toss mushroom caps with olive oil, kosher salt, and black pepper. Arrange gill side down and roast 6–8 minutes to release excess moisture. Drain any liquid from the pan and flip caps upright to cool slightly.
While the caps roast, melt 2 tbsp butter in a skillet over medium heat. Add chopped mushroom stems, onion, celery, and red pepper; cook 4–5 minutes until softened and most moisture evaporates. Stir in garlic and cook 30 seconds more.
Remove the skillet from heat. Stir in Old Bay, paprika, lemon zest, and parsley. Let cool 2 minutes, then mix in the Gouda so it starts to melt. Gently fold in the crab to keep the lumps intact; taste and add a pinch more salt only if needed.
For the topping, melt 1 tbsp butter in a small skillet over medium heat. Add panko and 1/8 tsp Old Bay; stir 2–3 minutes until lightly golden, then transfer to a bowl to stop the cooking.
Fill each mushroom cap with a heaping spoonful of the crab mixture and mound slightly. Sprinkle evenly with the toasted panko (and a few extra shreds of Gouda if you like). Tip: A small cookie scoop or piping bag makes neat, even portions.
Bake 8–10 minutes at 375°F until the centers are hot and the cheese is bubbling. For deeper color, broil 30–60 seconds at the end. Serve immediately with a little extra parsley or a quick squeeze of lemon.