Subbing in barley for refined Arborio rice adds both an appealing nutty flavor and extra nutrition. If you prefer not to cook with wine, substitute a mixture of 1/2 cup vegetable stock and 1 tablespoon lemon juice.
Bring broth to a boil in a saucepan over high. Reduce heat to low to keep warm.
Heat oil in a large Dutch oven over medium-high. Add mushrooms; cook, stirring occasionally, until softened, 5 to 6 minutes. Add onion, carrots, and garlic; cook, stirring often, until onions are translucent and soft, 5 to 6 minutes. Add barley; cook, stirring constantly, until toasted, 1 to 2 minutes.
Stir wine into barley mixture; let come to a boil over medium-high. Boil, stirring occasionally, until liquid is absorbed, 1 to 2 minutes. Stir in salt. Pour in enough of the warm broth (about 3 cups) to cover mixture. Let come to a boil; reduce heat to medium, and cook, stirring often, until liquid is absorbed, 5 to 6 minutes. Cook, stirring very often, until barley is tender, adding remaining broth 1 cup at a time, waiting until most of the broth is absorbed between additions, about 50 to 55 minutes total.
Stir asparagus into barley mixture; cook over medium, stirring occasionally, until barley is very soft and asparagus is crisp-tender, 5 to 8 minutes. Remove from heat. Stir in cheese, tarragon, and pepper.