Cut asparagus just below the tips, then on an angle at 1/4-inch intervals, discarding the woody part at the bottom. Set aside.
Bring water to a simmer and add 2 teaspoons of the salt. Keep on a low simmer.
Heat 2 tablespoons of the butter in a large skillet over medium-high heat and add mushrooms. Cook, stirring occasionally, until crispy and browned. Season with 1/2 teaspoon salt and pepper, to taste, and set aside.
In a large saucepan, add another 2 tablespoons of the butter over medium-high heat and let brown slightly until nutty. Add the shallot to the pot and cook until translucent, about 2 minutes. Add the rice and stir so that it is coated with the butter and glossy, about 1 minute. Stir in the remaining 2 1/2 teaspoons salt. Add the vermouth and cook, stirring constantly with a wooden spoon, until it is absorbed by the rice. Add the thyme springs. Ladle in about 1/2 cup of the simmering water and stir constantly, until the rice again absorbs the liquid, adjusting the heat to maintain a gentle simmer. Continue ladling in about 1/2 cup of water at a time, stirring between additions and letting the rice absorb the liquid before adding more.
When rice has absorbed about half the water (about 10 minutes into cooking process), stir in lemon zest. Continue stirring and adding water, 5 minutes more. Add asparagus. When rice is tender but al dente, after 20 or so minutes of cooking time, stop adding water. Vigorously beat in the remaining butter and cheese. Add the mushrooms, peas and lemon juice and stir just until heated through. Remove from heat. Let risotto rest for a minute or so before serving. Divide among 4 warm bowls, grind a generous amount of pepper over each, and serve.
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