In a large pot or Dutch oven, melt the butter over medium heat. Add the finely chopped onion and cook until it turns translucent, approximately 3-4 minutes. Be cautious not to burn the garlic, so add the minced garlic and sauté for an additional 1-2 minutes.

Sprinkle the all-purpose flour over the onions and garlic, stirring constantly to create a roux. Allow it to cook for about 1 minute to eliminate the raw flour taste.

While continuously stirring to prevent lumps, pour in the vegetable or chicken broth. Bring the mixture to a gentle boil.

Introduce the cubed potatoes to the pot and season with salt and pepper to your liking. Reduce the heat to medium-low, cover the pot, and let the soup simmer for around 15-20 minutes or until the potatoes turn tender.

Once the potatoes are fully cooked, use a potato masher or a spoon to slightly mash some of them in the soup. This technique will help thicken the soup and give it a creamier consistency.

Incorporate the heavy cream or half-and-half into the mixture, allowing the soup to simmer for an extra 5-10 minutes. This step will enable the flavors to blend harmoniously.

Taste the soup and adjust the seasoning as necessary with more salt and pepper.

Remove the pot from the heat. For a smoother texture, use an immersion blender to blend the soup according to your preferred consistency. Alternatively, leave it chunky for a more textured experience.

Serve the Creamy Potato Soup in bowls and garnish with your chosen toppings, such as grated cheese, chopped chives, crispy bacon, or a dollop of sour cream.