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1
Make the polenta: Whisk together cornmeal, salt and water in a medium saucepan. Bring to a boil over medium-high heat. Whisk in the milk. Reduce the heat to a simmer, cover and cook, stirring occasionally until polenta is tender and thickened, about 45 minutes. Vigorously stir in the butter until melted. Turn off the heat. Stir in the cheese and pepper to taste. Add salt or adjust as needed.
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2
In the meantime, prepare the asparagus: Wash the spears, trim the ends off and cut the stalks into 1-inch pieces. Dry thoroughly and spread out on a sheet pan. Preheat the broiler until hot, about 5 minutes. Broil the asparagus for 5-10 minutes until tender and beginning to brown in spots. Set aside.
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3
Poach the eggs: Bring several inches of water to a boil in a large saucepan. Reduce the heat so that the water barely bubbles. Crack the eggs into individual measuring cups or ramekins. Gently tip each egg into the water and use a spoon to pull the white up and over the yolk to help it all stay together. Set a timer for 3:40 (yolks will still be runny; go longer if you like a firmer yolk) and fill a bowl with hot water. Remove the eggs from the water with a slotted spoon and transfer to the hot water to keep warm.
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4
Make the sauce: Heat a frying pan over medium heat. Cut the butter into equal-sized pieces and add to the pan. Stir and swirl the pan until the butter melts, foams and begins to brown. Stir in lemon juice and salt and pepper to taste. The sauce will bubble. Tip in the asparagus and cook another minute or two until hot and slightly darkened in color. Turn off the heat and stir in all but one large pinch of lemon zest.
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5
To serve: Divide the polenta between two bowls. Use a slotted spoon to remove the asparagus from the frying pan and add it to the top of the polenta. Again using a slotted spoon, remove the eggs from the hot water, letting the water drain off. You can also blot them with a paper towel. Place one egg on each bowl. Tilt the frying pan and drizzle or spoon the brown butter left in the pan over each bowl. Top each bowl with a pinch of lemon zest, additional Parmesan cheese and black pepper. Serve immediately.
