In a large Dutch oven or heavy-bottomed pot, cook the chopped bacon over medium heat until crisp, about 8-10 minutes. Use a slotted spoon to transfer the bacon to a paper towel-lined plate, leaving the drippings in the pot.
Add the olive oil to the bacon drippings in the pot. Add the chopped onion, carrots, and celery, and cook over medium heat, stirring occasionally, until the vegetables have softened, about 5-7 minutes. Stir in the minced garlic and cook for 1 minute more until fragrant.
Sprinkle the flour over the vegetable mixture and cook, stirring constantly, for 1 minute to form a paste-like roux and cook off the raw flour taste. Gradually pour in the chicken broth while whisking continuously to avoid lumps.
Bring the soup to a simmer, then stir in the gnocchi. Cook for 3-4 minutes, or until the gnocchi are tender and begin to float to the surface.
Reduce the heat to low and slowly stir in the half-and-half, ensuring the soup does not boil. Add the shredded cheddar cheese one handful at a time, stirring until completely melted and the soup is smooth.
Stir in the reserved cooked bacon, salt, and pepper. Taste and adjust seasoning if needed. Ladle the soup into bowls and garnish with fresh parsley before serving.