Broccoli, Cheddar, and Brown Rice Cakes

Cooking spray

unsalted butter

chopped yellow onion

garlic cloves, chopped

unsalted vegetable stock (such as Swanson)

fresh broccoli florets, cut into 1/2-in. pieces

pkg. precooked brown rice (such as Uncle Ben's)

whole-wheat panko (Japanese breadcrumbs)

grainy mustard

black pepper

kosher salt

preshredded reduced-fat sharp cheddar cheese, divided (about 3/4 cup)

large eggs, lightly beaten

Sliced green onions (optional)