Place a steamer insert into a saucepan and fill with water to just below the bottom of the steamer. Bring water to a boil. Add broccoli and green beans, cover, and steam until tender, 2 to 6 minutes.

Preheat oven to 350 degrees F (175 degrees C).

Place broccoli and green beans in a 9x13-inch baking dish. Cover with chicken and 1 1/2 cups Cheddar cheese.

Heat chicken broth in a saucepan over medium heat until almost boiling. Mix a small amount of broth with flour in a bowl to create a paste; whisk paste into remaining chicken broth until thickened. Stir in yogurt, fish sauce, lime juice, and curry powder. Season with salt and pepper. Pour curry mixture over chicken. Sprinkle with remaining 1/2 cup Cheddar cheese.

Bake in preheated oven until bubbly, 30 to 45 minutes.