Shirred Eggs with Vegetable Hash and Smoked Cheddar

In this single-skillet supper, eggs are cracked over a tender sweet potato and zucchini hash, which is then topped with smoked Cheddar and broiled.

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  2 medium sweet potatoes, peeled and cut into 1-inch pieces
  
  


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  2 medium carrots, peeled and cut into 1-1/2-inch pieces
  
  


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  2 medium zucchini, cut into 1-1/2-inch pieces
  
  


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  3 Tbs. olive oil
  
  


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  1 Tbs. unsalted butter
  
  


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  1 Tbs. finely chopped fresh sage
  
  


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  1/4 tsp. cayenne; more to taste
  
  


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  Kosher salt
  
  


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  8 large eggs, at room temperature
  
  


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  Freshly ground pepper
  
  


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  4 oz. smoked Cheddar, coarsely grated (about 1-1/2 cups)
  
  

Position a rack 6 inches from the broiler and heat the broiler on high.

Working in two batches, pulse the sweet potatoes, carrots, and zucchini in a food processor until chopped, with pieces no larger than almonds in the mix. Transfer to a bowl.

Heat the oil and butter in a 12-inch cast-iron skillet over medium-high heat until the butter melts and the foam subsides. Add the vegetables and the sage and cook, stirring often, until tender and golden-brown in spots, about 10 minutes. Remove from the heat and season with the cayenne and 1-1/2 tsp. salt.

Smooth the surface of the hash and crack the eggs over it, leaving a little space between each egg. Season each egg with salt and pepper and broil until the whites are set, 1 to 2 minutes. Sprinkle the Cheddar evenly over the eggs and hash and broil until the cheese is melted, about 1 minute. Let cool slightly before serving.