Heat a large soup pot or Dutch oven over medium-high heat and brown the ground beef, breaking it into small crumbles as it cooks, about 5-7 minutes. Drain excess fat and set the cooked beef aside on a plate. In the same pot, melt 1 tablespoon of butter over medium heat, then add the diced onion, shredded carrots, dried parsley, and dried basil. Sauté for 4-5 minutes until the vegetables soften and release their aromatics—this builds the flavor foundation of the soup.
Pour the chicken broth into the pot with the softened aromatics, then add the diced potatoes and broccoli florets. Bring the mixture to a boil over medium-high heat, then reduce to medium and simmer covered for 10-12 minutes until the potatoes are tender when pierced with a fork. The vegetables need time to soften before the cheese and cream are added, which would lower the heat and prevent further cooking.
While the soup simmers, melt 3 tablespoons of butter in a separate skillet over medium heat. Once melted and foamy, sprinkle in the flour and stir continuously for 3-5 minutes until the mixture turns light brown and bubbles slightly—this cooks out the raw flour taste and creates a roux that will thicken the soup. I like Kerrygold butter here because its richness adds subtle flavor to the final dish.
Once the potatoes are tender, slowly stir the roux from Step 3 into the simmering soup, whisking to avoid lumps. Bring the soup back to a gentle boil and cook for 2 minutes to fully activate the thickening. Reduce heat to low and stir in the cooked ground beef from Step 1, sharp cheddar cheese, milk, Worcestershire sauce, salt, pepper, and garlic powder. Stir gently and continuously until the cheese completely melts, about 3-5 minutes—the low heat prevents the soup from breaking or becoming grainy.
Remove the soup from heat and stir in the sour cream until fully incorporated. I always use Daisy brand for a creamy, tangy finish that rounds out the rich cheese flavors beautifully. Taste and adjust seasoning if needed with additional salt and pepper. Ladle into bowls and serve hot.