Step 1: In a large pot or Dutch oven, melt the unsalted butter over medium heat.
Step 2: Add the finely diced yellow onion, celery, and carrots. Sauté for 5-7 minutes, or until the vegetables begin to soften.
Step 3: Stir in the minced garlic and cook for another minute until fragrant.
Step 4: Sprinkle the all-purpose flour over the sautéed vegetables. Stir constantly for 1-2 minutes to create a roux, cooking out the raw flour taste.
Step 5: Gradually whisk in the chicken broth, a little at a time, to prevent lumps. Continue whisking until the mixture is smooth and slightly thickened.
Step 6: Add the peeled and diced Yukon Gold potatoes to the pot. Bring the soup to a gentle simmer, then reduce heat to low, cover, and cook for 15-20 minutes, or until the potatoes are fork-tender.
Step 7: Stir in the half-and-half. Heat through gently, but do not bring the soup to a rolling boil after adding the half-and-half to prevent curdling.
Step 8: Season the soup with salt and black pepper to taste. Adjust as needed.
Step 9: Ladle the warm soup into bowls. Garnish with fresh chives or parsley, if desired, before serving.