Preheat oven to 450°F. Coat a baking sheet with cooking spray.
Melt butter in a large skillet over medium-high. Add onion and garlic; sauté 4 minutes. Add stock and broccoli. Bring to a boil; cook 3 minutes.
Heat rice according to package directions. Combine broccoli mixture, rice, panko, mustard, pepper, salt, and 1/2 cup cheese in a large bowl. Stir in eggs. Divide and shape broccoli mixture into 8 (2 1/2-inch) patties. Arrange patties on prepared pan; coat patties with cooking spray. Bake at 450°F for 15 minutes. Top with remaining 1/4 cup cheese, and bake at 450°F for 4 more minutes or until cheese melts. Garnish with green onions, if desired.