Step 1

Place broth in a medium saucepan; bring to a simmer over medium-high heat. Reduce the heat so the broth remains steaming, but is not simmering.

Step 2

Meanwhile, heat oil in a large saucepan over medium heat. Add shallots; cook, stirring, until fragrant, about 1 minute. Stir in squash and mushrooms; cook, stirring often, until the mushrooms give off their liquid, about 5 minutes. Add thyme, salt, pepper and saffron (if using); cook for 30 seconds. Add rice; stir until translucent, about 1 minute. Add wine (or vermouth) and cook, stirring, until almost absorbed by the rice, about 1 minute.

Step 3

Stir in 1/2 cup of the hot broth; reduce heat to a gentle simmer and cook, stirring constantly, until the liquid has been absorbed. Continue adding the broth 1/2 cup at a time, stirring after each addition until all the liquid has been absorbed, until the rice is tender and creamy, 30 to 40 minutes total. (You may have some broth left.) Remove from the heat and stir in cheese.