Combine broccoli, onions, and mushrooms in greased 10-inch microproof quiche dish or pie plate. Cover and cook on HIGH (maximum power) 5 minutes.
Blend eggs and half and half in medium bowl. Add rice, cheeses, salt, and pepper; stir. Pour mixture over vegetables; stir gently. Cook uncovered on MEDIUM (50% power) 9 to 11 minutes, or until quiche is set. Let stand 10 minutes.
In a mixing bowl, whisk the cream and the egg yolks together. Whisk well. Season with salt and pepper. Butter a square baking dish. Lay a layer of the potatoes over the bottom of the pan. Season the layer with salt and pepper.
Crumble an eighth of the cheese over the potatoes. Ladle 1 cup of the cream mixture over the cheese. Lay another layer of potatoes over the cream mixture. Using your hands, firmly press the layers together. Repeat the process with the remaining ingredients.
Place a piece of parchment paper over the potatoes and cover with foil. Place in the oven and cook for 30 minutes. Remove the foil and continue to cook for 15 minutes. Remove from the oven and cool completely. Reheat and slice into individual servings.