Melt butter in a large Dutch oven or pot over medium heat. Add diced onion and celery and cook until softened, about 5-7 minutes. Stir in the minced garlic and cook for 1 more minute until fragrant.

Sprinkle the flour over the vegetables and stir constantly for 1 minute to cook out the raw flour taste.

Gradually whisk in the chicken broth until smooth. Add the cubed potatoes and bring to a simmer. Reduce heat to low, cover, and cook for 15-20 minutes, or until potatoes are very tender.

Use an immersion blender to partially blend the soup until it reaches your desired consistency, leaving some chunks of potato for texture. Alternatively, transfer 3 cups to a blender, blend until smooth, and return to the pot.

Stir in the milk and heavy cream and warm through over low heat. Do not let the soup boil.

Remove the pot from the heat. Gradually stir in the shredded cheddar cheese until completely melted and smooth.

Season with salt and pepper to taste. Serve hot, garnished with optional toppings like bacon, sour cream, and chives.