Preheat oven to 400°F. Sprinkle mushroom caps with 1/4 teaspoon salt and place rounded side up on an oiled baking sheet. Bake until just barely tender, about 15 minutes.
Meanwhile, combine greens, onion, garlic, pepper, remaining 1/4 teaspoon salt and 1/4 cup water in a large skillet. Bring to a boil over medium-high heat; cover and cook, stirring frequently, until greens are very tender, 8 to 12 minutes. Add a few tablespoons water if the skillet starts to dry out.
Turn mushrooms over and fill caps with greens. Sprinkle with blue cheese, return to the oven and bake until cheese is melted, about 5 minutes.
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