In a large pot, over medium heat, add the butter and olive oil. Add the onions and cook for a minute.
Then add the broccoli, carrots, garlic, salt, pepper, and red pepper flakes.
Cook and stir for about 3 minutes to lightly softened the vegetables.
Add the flour into the softened vegetables and cook and stir for about a minute.
Add the chicken broth and mix well. Next, add the milk and the half 'n half.
Bring the broccoli soup to a simmer to warm through. Taste and adjust seasoning if necessary.
Finally, add the shredded cheddar cheese and stir until the cheese has completely melted.
Notes
Store leftovers airtight in the fridge up to 3 days; reheat gently, stirring often.Frozen broccoli works in place of fresh, add it straight to the pot while frozen. For a smoother texture, blend part or all of the soup with an immersion blender, or carefully blend in batches in a countertop blender. Don't overfill, since hot soup can build pressure and push the lid off.