Preheat the oven to 450 degrees F. Lightly spray a 6-cup muffin tin with nonstick cooking spray.
Place the thawed hash browns in a large bowl. Add the butter, thyme, 2 egg whites, chopped provolone and 1/2 teaspoon salt. Whisk thoroughly to combine. Evenly divide the potato mixture among the prepared muffin cups. Firmly press the potatoes into the bottoms and up the sides of each cup. Bake until golden brown and crispy, 15 to 18 minutes.
Place the zucchini rounds into the bottoms of each baked cup. Cover each with 2 pieces of provolone. Bake until the zucchini is softened and the provolone melts, an additional 3 minutes. Let cool slightly.
Crack a whole egg into a small bowl, then add it to the top of a baked cup. Repeat with the remaining eggs. Top each cup with mushroom slices. Sprinkle with salt and pepper and bake until the egg whites are set and the yolks are still runny, 12 minutes more. Loosen from cups with a small knife and serve immediately.
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