Preparation Heat 2 tablespoons oil in extra-large skillet over high heat. Add mushrooms; sprinkle with coarse salt and pepper. Sauté until brown, about 10 minutes. Transfer to bowl. Pour 1/4 cup broth into same skillet. Add 1/3 of greens. Sprinkle with coarse salt and pepper. Toss until wilted but still bright green, about 2 minutes. Transfer to large strainer set over bowl. Repeat 2 more times with remaining broth and greens. DO AHEAD: Can be made 2 hours ahead. Let mushrooms and greens stand at room temperature. Reserve skillet. Heat 2 tablespoons oil in same skillet over medium-high heat. Add onions. Sauté until beginning to color, about 5 minutes. Add garlic and crushed red pepper; stir 1 minute. Add greens and mushrooms. Toss to heat through, about 2 minutes. Season with salt and pepper. Transfer to bowl.
Preparation Heat 2 tablespoons oil in extra-large skillet over high heat. Add mushrooms; sprinkle with coarse salt and pepper. Sauté until brown, about 10 minutes. Transfer to bowl. Pour 1/4 cup broth into same skillet. Add 1/3 of greens. Sprinkle with coarse salt and pepper. Toss until wilted but still bright green, about 2 minutes. Transfer to large strainer set over bowl. Repeat 2 more times with remaining broth and greens. DO AHEAD: Can be made 2 hours ahead. Let mushrooms and greens stand at room temperature. Reserve skillet. Heat 2 tablespoons oil in same skillet over medium-high heat. Add onions. Sauté until beginning to color, about 5 minutes. Add garlic and crushed red pepper; stir 1 minute. Add greens and mushrooms. Toss to heat through, about 2 minutes. Season with salt and pepper. Transfer to bowl.