Preheat oven to 350°F with a rack in top third position (or 6 inches from heat source). Melt 3 tablespoons butter in a 10-inch cast-iron or other deep ovenproof skillet over medium-high heat. Add sliced mushrooms and ¼ teaspoon salt; cook, stirring occasionally, until browned, about 16 minutes. Transfer to a plate; do not wipe the skillet clean.

Melt the remaining 3 tablespoons butter in the skillet over medium heat. Add sliced leeks and ¼ teaspoon salt; cook, stirring often, until softened and slightly translucent, about 5 minutes. Add chopped garlic, 2 teaspoons each rosemary and thyme and ½ teaspoon pepper; cook, stirring constantly, until fragrant, about 1 minute.

Add 3 tablespoons flour; cook, stirring constantly, for 1 minute, then gradually add 2½ cups milk (about ¾ cup at a time); cook, stirring often, until the sauce thickens and coats the back of a spoon, 4 to 5 minutes. Stir in ¼ teaspoon nutmeg and the remaining ¼ teaspoon salt.

Add drained spinach and 1 (5.2-ounce) package garlic-and herb cheese; cook over medium-low heat, stirring constantly, until the cheese is melted and the spinach is evenly distributed, about 2 minutes. Remove from heat; fold in prepared rice and the cooked mushrooms until combined.

Spread into an even layer; sprinkle with 1 cup Gruyère. Bake until the cheese is melted, 10 to 12 minutes. Increase oven temperature to broil (do not remove skillet from oven) and broil until the cheese is browned, 2 to 3 minutes. Let rest for 10 minutes before serving; garnish with additional thyme, if desired.