Melt butter in a large pot or Dutch oven over medium heat. Add chopped onion and cook until softened, about 5 minutes.
Stir in minced garlic and cook for 1 minute more until fragrant.
Add quartered new potatoes, green beans, diced ham, chicken broth, black pepper, and salt to the pot.
Bring the mixture to a boil, then reduce heat to low, cover, and simmer for 20-25 minutes, or until potatoes are tender and green beans are cooked through.
Stir gently before serving.
