Preheat your oven to 375°F (190°C).
In a medium saucepan, bring water to a boil, then add the trimmed green beans. Blanch them for about 3 minutes until they are bright green but still crisp. Drain and set aside.
In a large mixing bowl, combine sliced potatoes, chopped onion, minced garlic, heavy cream, salt, pepper, paprika, and thyme. Mix until all ingredients are well combined.
Gently fold the blanched green beans into the potato mixture.
Pour the mixture into a greased 9x13-inch baking dish, spreading it evenly.
In a skillet, melt the butter over medium heat. Add the breadcrumbs and stir until they are toasted and golden brown.
Remove the skillet from the heat and add half of the shredded cheddar cheese to the breadcrumbs, mixing well.
Sprinkle the breadcrumb and cheese mixture evenly over the potato and green bean mixture in the baking dish.
Top with the remaining shredded cheddar cheese.
Cover the baking dish with aluminum foil and bake in the preheated oven for 25 minutes.
After 25 minutes, remove the foil and bake for an additional 15 minutes, or until the top is golden and the potatoes are fork-tender.
Allow the dish to cool slightly before serving.