Place the trimmed and cut green beans into the bottom of a 6-quart slow cooker.
Melt the butter in a large skillet over medium heat, then add the chopped onion and cook for 4-5 minutes until softened and translucent.
Add the sliced mushrooms to the skillet and cook for 6-7 minutes until they have released their liquid and begun to brown, then stir in the minced garlic, salt, pepper, and paprika and cook for 1 more minute until fragrant.
Sprinkle the flour over the mushroom mixture and cook for 1 minute while stirring constantly, then slowly pour in the milk and chicken broth while whisking continuously until the sauce is smooth and begins to thicken, about 3-4 minutes.
Remove the skillet from the heat and stir in the grated cheddar cheese until fully melted and the sauce is smooth.
Pour the creamy mushroom sauce evenly over the green beans in the slow cooker and stir gently to combine all ingredients.
Cover and cook on LOW for 5-6 hours or on HIGH for 3-4 hours, until the green beans are tender and the sauce is bubbly.
Sprinkle the French-fried onions evenly over the top of the casserole, cover again, and cook for an additional 15 minutes on HIGH until the onions are warmed and slightly softened.
