Cook the bacon in a large, deep skillet over medium-high heat until evenly browned, about 10 minutes. Drain on a paper towel-lined plate. Crumble and set aside.

Meanwhile, melt 2 tablespoons of butter in a large pot over medium heat. Stir in the chopped onion and whole garlic clove. Cook and stir until the garlic has softened, and the onions begin to turn golden-brown, about 8 minutes. Scrape the mixture into a small bowl; remove the garlic clove, mash, and return to the onions. Place the green beans, wax beans, and carrots into the saucepan. Pour in the chicken broth and enough water to just cover the vegetables; season with salt. Bring to a boil over high heat; reduce heat to medium-low and simmer 15 minutes.

After the beans have simmered 15 minutes, stir in the new potatoes and cook until tender, about 30 minutes more. Stir in the garlic, onions, 1/4 cup butter, and cream; return to a simmer. Dissolve the flour in 1/2 cup of water and stir into the soup. Cook and stir until the soup has thickened, 10 to 15 minutes. Serve with crumbled bacon.