Preheat the oven to 400 degrees F. Trim the woody ends of the asparagus and break the remaining stalks into 2-3 pieces. Toss with the olive oil and salt to taste. Lay in a baking dish and roast, tossing occasionally, about 20 minutes.
Meanwhile, heat 1 tablespoon of the butter in a large saucepan with high sides over medium heat. Add the peeled, smashed clove of garlic. Cook, stirring occasionally, until the garlic begins to turn golden. Remove and discard. Add the peas and salt to taste. Cook 2-3 minutes, until the peas are just cooked through. Use a skimmer to remove the peas, leaving the butter behind. Set to the side.
Add the shallots to the butter remaining in the pan. Cook, stirring occasionally, until translucent. Add the rice and cook until translucent. Deglaze the pan with the white wine, cooked asparagus and their juices, as well as the reserved peas to the risotto. Stir to combine.
Cover the pot and set aside. In a frying pan, heat the oil and crack 4 eggs. Season with salt and pepper. Cook until the white is cooked through, about 4 minutes, or until your preferred doneness. Add the reserved butter, Parmesan and basil to the risotto. Season to taste with salt and pepper. Serve immediately in soup plates, topped with an egg.