Clean mushrooms with a damp paper towel, remove stem, and set aside. Using a teaspoon, gently scrape the dark gills to remove from under the cap of the mushroom. Discard gills.
Coarsely chop the stems for the stuffing, discarding any hard edges. Brush mushroom generously with olive oil on the inside and outside; set aside until stuffing is ready.
Saute onion and celery in 2 tablespoons of olive oil in a skillet over medium heat until vegetables are softened and beginning to brown, about 10 minutes.
Add the chopped mushroom stems and garlic to the skillet and continue to cook 3-4 minutes or until garlic is fragrant. Transfer onion, celery, and garlic from the skillet to a medium sized mixing bowl.
Add zucchini and red pepper to the same skillet with 2 tablespoons olive oil and cook over medium heat until vegetables soften and release extra moisture, about 5 minutes.
Sprinkle with Italian seasoning, salt, and pepper and cook for 2-3 minutes to meld flavors. Transfer to the mixing bowl with the onions and celery, stir to combine. Allow mixture to cool slightly while grill is preheating.
Just before grilling, toss blue cheese into vegetable stuffing then salt and pepper, to taste. Spoon evenly into the mushrooms. Top with additional blue cheese and grill over medium-low heat or in a preheated 350° oven for 10 minutes.
Mushrooms will soften and wilt slightly when done. Use a spatula to remove from the grill and serve immediately or transfer to an oven safe dish to keep warm in a 200°F oven until ready to serve.
Make-ahead up to the point of baking and store in the fridge for up to 3 days