Sauté the base: Warm the olive oil in a large pot or Dutch oven over medium heat. Add the chopped onion and frozen broccoli and cook for 5–10 minutes, stirring occasionally, until softened and lightly golden. Add the garlic and cook for about 30 seconds, just until fragrant.
Simmer the soup: Pour in the chicken broth. Bring to a gentle boil, then reduce the heat to low. Cover and simmer for 15–20 minutes, until the broccoli is very tender.
Blend until smooth: Use an immersion blender to blend the soup directly in the pot until smooth and creamy. If using a blender, work in batches and return the soup to the pot.
Finish with cream: Stir in the heavy cream and season with salt and pepper to taste. Keep the heat low and let the soup warm through for about 5 minutes.
Serve simply: Ladle into bowls and top with shredded cheddar cheese. Serve warm with sourdough or in a crusty bread bowl.