Put the broth in a 3-quart saucepan, add a pinch of salt, and bring to a boil over high heat; lower the heat to maintain a simmer.
Heat the oil in a wide, heavy-duty 5- to 6-quart pot over medium-high heat. Add the shallots, lower the heat to medium, and cook, stirring occasionally, until softened and translucent, about 3 minutes. Add the garlic and cook, stirring, until softened and fragrant, about 1 minute. Add the mushrooms, raise the heat to high, and cook, stirring frequently, until softened, about 2 minutes. Lower the heat to medium, add the rice, and cook, stirring often, until the rice is lightly toasted, about 3 minutes.
Add the wine and cook, stirring, until most of it is absorbed, about 30 seconds.
Stir about 1-1/2 cups of the simmering broth into the rice. Lower the heat to maintain a simmer and cook, stirring frequently, until most of the broth is absorbed, about 1 minute. Add another 1-1/2 cups broth and cook, stirring frequently, until most of it is absorbed, about 3 minutes. Repeat the process once or twice more, tasting the rice every few minutes after the third broth addition until it’s just shy of firm to the bite but without a crunchy center, about 12 minutes after the first addition of broth.
Stir in the asparagus and 1 cup broth. Cover, lower the heat to the low, and cook until the asparagus is crisp-tender and the rice is tender but with some resistance, about 5 minutes. Remove from the heat and fold in the cheese. Cover and let stand for 5 minutes. Season to taste with salt and serve right away, sprinkled with black pepper.