Pour oil into a 5- to 6-quart pan over medium-high heat; when hot, add onion and mushrooms and stir often until onion is limp and liquid has evaporated, 5 to 6 minutes. Add rice and stir until opaque, about 3 minutes.
Add wine and stir until it's absorbed, 1 to 2 minutes. Add 4 1/2 cups broth and bring to a boil over high heat, stirring often. Reduce heat and simmer, stirring often until rice is tender to bite, 15 to 20 minutes. If a creamier consistency is desired, stir in about 1/2 cup more broth. Stir in cheese, parsley, sage, tarragon, and thyme. Add salt and pepper to taste.
Cleaning mushrooms: Trim off tough or discolored bottoms of mushroom stems and any bruised spots or blemishes. (For shiitakes and oysters, remove the entire fibrous stem.) For firm mushrooms such as portabellas, wipe dirt off with a damp cloth or place in a colander, rinse thoroughly under cool running water, and pat dry with towels. For delicate mushrooms that have lots of places for dirt to hide, such as chanterelles and hedgehogs, submerge in a bowl of cool water and gently agitate with you hands to loose any particles. Drain, rinse carefully under running water, and gently pat dry with a towel.