Pour 2 cups boiling water into a large bowl. Add dried mushrooms; let stand until rehydrated, about 20 minutes. Remove mushrooms with a slotted spoon and pat dry; chop into smaller pieces. Reserve soaking liquid.

Pour chicken broth into a saucepan over medium-low heat. Cover and keep warm.

Process cauliflower in a food processor fitted with a shredding disc into grains the size of rice.

Melt butter in a large pot over medium heat. Add shallot; cook until softened, about 2 minutes. Add cauliflower rice, salt, and pepper. Cook, stirring occasionally, until flavors combine, about 5 minutes.

Stir 1 cup of warm chicken broth into the pot; simmer until absorbed, about 5 minutes. Add remaining broth, 1 cup at a time, simmering until it is absorbed and cauliflower rice is tender, about 10 minutes.

Stir chopped mushrooms, peas, balsamic vinegar, and thyme into the pot. Cook and stir until mushrooms and peas are heated through, about 5 minutes.