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Preheat the oven to 400 degrees.
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On a baking sheet lined with foil, arrange the vegetables in a single layer, toss with olive oil and generously salt and pepper. Roast until soft and slightly browned, about 30-40 minutes. Set aside and leave the oven on.
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In an oven-proof skillet, cook the chorizo. Remove from the pan and drain. Set aside.
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In the same skillet, warm the milk and broth. Add the cornmeal and whisk to combine, then let it come to a gentle boil, stirring frequently. Add the cheese, stir to combine, then salt and pepper to taste.
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Sprinkle the spinach over the top of the polenta, then layer on the vegetables and finally the chorizo. With a spoon, make four wells and crack the eggs into them. Put the skillet in the oven and cook for about 10-15 minutes, until the egg whites are opaque.
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Serve immediately.