Heat 2 tablespoons of extra-virgin olive oil in a Dutch oven or stockpot over medium heat. Add the sausage, cook until browned, then transfer to a plate and set aside.

Add the onion, bell pepper, and celery to the same pan and cook for 2 to 3 minutes until softened. Add the minced garlic and cook for another minute.

Add the butter and flour to the pan with the vegetables and stir with a wooden spoon to make a paste. Cook for 2 to 3 minutes to eliminate the raw flour taste.

While constantly stirring, slowly add the broth and half-and-half. Add the potatoes and seasonings, lower the heat to medium-low, and simmer for 20 to 25 minutes until the potatoes are fork-tender, stirring occasionally.

Add the cooked sausage and cheddar cheese to the pot and stir until the cheese is melted. Remove the soup from the heat, cool slightly, and ladle into bowls.