Begin by rinsing the rice under cold water to remove excess starch. Combine the rinsed rice with vegetable broth, adding salt and black pepper to taste. Bring to a boil over medium-high heat, then reduce heat to low and cover. Simmer for 18 minutes, fluffing rice with a fork once cooked.

Heat olive oil in a skillet over medium heat. Add diced onion and sauté for about 5 minutes until translucent. Stir in minced garlic and cook for an additional minute.

Add chopped mushrooms, thyme, and sage to the skillet with the onions and garlic. Sauté for 8 to 10 minutes until mushrooms are softened and have released their juices.

In a separate saucepan, melt butter over medium heat. Whisk in flour for 1-2 minutes to form a roux. Gradually pour in milk, whisking continuously until thickened, about 5 to 7 minutes.

Add salt, pepper, and nutmeg to the béchamel sauce. Stir in Parmesan cheese until melted and fully incorporated.

Preheat oven to 350°F (175°C). In a large mixing bowl, combine rice, mushroom mixture, and béchamel sauce. Fold until evenly coated.

Transfer to a greased 9x13-inch baking dish, spreading evenly. Bake for 25 to 30 minutes until golden and bubbly.

Let the casserole rest for 5 to 10 minutes before serving to allow flavors to meld.