Preheat oven to 375°F and lightly grease a 9x13-inch baking dish with butter or nonstick spray.

Melt butter in a skillet over medium heat, add chopped onion and minced garlic, cooking until fragrant and translucent, about 3 minutes.

Sprinkle flour over the aromatics, then whisk in chicken broth and cream until smooth. Simmer until the sauce thickens, about 4 minutes.

Stir in green beans and shrimp, season with salt and pepper. Cook until shrimp turns pink and beans are bright green, about 5 minutes.

Pour the shrimp and bean mixture into the prepared dish, spreading it evenly.

Combine breadcrumbs and grated Parmesan in a small bowl, tossing to coat crumbs.

Sprinkle the breadcrumb mixture evenly over the casserole.

Bake at 375°F for 15–20 minutes, until the topping is golden and the casserole is bubbly at the edges.

Let the casserole cool for 5 minutes before serving.