In a large pot over medium heat, cook bacon until crisp, about 6–8 minutes; transfer to paper towels to drain.
Add olive oil to the rendered fat, add diced onion and sauté 4–5 minutes until translucent, then toss in minced garlic and cook 30 seconds until fragrant.
Sprinkle flour over the aromatics and stir 1–2 minutes to form a roux; gradually whisk in chicken broth, bring to a gentle simmer, and cook 5 minutes until slightly thickened, then stir in heavy cream.
Add gnocchi and simmer 3–4 minutes until they float; remove from heat, fold in shredded cheddar and reserved bacon, season with salt and pepper.
Garnish
Scatter chopped fresh chives on top before serving.
Notes
Cool soup within 2 hours at room temperature. Store in an airtight container up to 4 days in the fridge or freeze for up to 3 months (leave headspace). Reheat gently over medium-low heat, stirring occasionally, until it reaches 165°F for safety.For make-ahead: prepare through the roux stage, chill, then finish with cream, gnocchi, cheese, and bacon when ready to serve.