Melt 2 tablespoons butter in a stockpot over medium heat; cook and stir onion and garlic in the melted butter until tender, 8 to 10 minutes. Add potato and broccoli stems; generously season with salt and pepper. Pour broth into potato mixture, cover, and simmer until potatoes are softened, about 10 minutes.
Mix broccoli florets into soup and simmer until broccoli is tender, about 5 minutes. Blend soup with an immersion blender until smooth.
Melt 3 tablespoons butter in a small saucepan over medium heat; whisk flour, milk, and heavy cream into melted butter. Season with salt and pepper. Whisk milk mixture until bubbling and thickened, about 5 minutes. Whisk thickened milk mixture into soup until incorporated; adjust salt and pepper.
