Step 1

Preheat broiler and position a rack 6 to 8 inches from the heat source. Melt butter in a large high-sided ovenproof skillet over medium-high heat. Add mushrooms and onion; cook, stirring often, until browned and tender, about 7 minutes. Add garlic and thyme; cook, stirring constantly, until fragrant, about 1 minute. Add rice, stirring to coat and break up clumps.

Step 2

Whisk broth, sour cream, mayonnaise, mustard, salt and pepper in a bowl until smooth. Whisk in cornstarch. Add the mixture to the pan; stir to combine. Stir in broccoli; let the mixture come to a boil. Cover, reduce heat to medium and cook until the broccoli is bright green and tender, about 7 minutes. Remove from heat; sprinkle with cheese.

Step 3

Transfer the skillet to the oven; broil until the cheese has melted, about 2 minutes. Serve hot.