Sauté the aromatics. In a large pot over medium heat, melt the butter (or warm the oil). Add the onion and celery with a pinch of salt. Cook for 5–7 minutes, stirring occasionally, until soft and translucent. Add the garlic and cook for 30 seconds until fragrant.

Add the flour. Sprinkle the flour over the vegetables and stir for 1 minute. This creates a light roux that will help thicken the soup later.

Pour in the broth. Slowly whisk in the broth to avoid lumps. Add the potatoes, bay leaf, thyme, 1 teaspoon salt, and a few grinds of pepper. Bring to a gentle boil.

Simmer until tender. Reduce heat to medium-low and simmer uncovered for 15–20 minutes, stirring occasionally, until the potatoes are very tender but not falling apart.

Thicken and cream. Fish out the bay leaf. Using a potato masher, mash some of the potatoes directly in the pot—about 1/3 of them—for a thicker, creamier texture. Stir in the milk or half-and-half. If you prefer it richer, add the sour cream now.

Adjust seasoning. Taste and add more salt and pepper as needed. Stir in the final tablespoon of butter for extra silkiness.

Optional cheese finish. If using cheddar, remove the soup from the heat and stir it in until melted and smooth. Avoid boiling after adding cheese to prevent graininess.

Serve with toppings. Ladle into bowls and garnish with crumbled bacon, green onions or chives, extra cheese, and a crack of pepper.