Step 1

Preheat the oven to 425 degrees F. Pierce the potatoes in a few places with a fork and microwave until just tender, 6 to 8 minutes. Set aside to cool.

Step 2

Heat 2 tablespoons olive oil in a large ovenproof nonstick skillet over medium-high heat. Add the mushrooms, season with salt and pepper and cook, stirring, until browned, 5 minutes. Add the garlic and cook until just softened, about 1 minute. Whisk 3 eggs, 1 tablespoon olive oil, the flour, milk, 3/4 teaspoon salt and a few grinds of pepper in a medium bowl. Pour over the mushrooms and sprinkle with the parsley. Transfer to the oven and bake until puffed and browned around the edges, about 20 minutes.

Step 3

Meanwhile, combine the remaining 1 tablespoon olive oil and the chorizo in another large nonstick skillet; cook over medium heat, stirring occasionally, until beginning to crisp, about 5 minutes. Chop the potatoes into bite-size pieces and add to the skillet; season with salt and pepper. Cook until the potatoes are slightly crisp, 8 to 10 minutes.

Step 4

Make 4 openings in the potato mixture and crack an egg into each spot; season with salt and pepper. Cook until the whites are set but the yolks are still soft, 3 to 5 minutes. Cut the Dutch baby into wedges. Serve with the potatoes, chorizo and eggs; sprinkle with parsley.