Preheat your oven to 400°F (200°C). Toss the broccoli florets with oil, salt, and pepper. Spread them in a single layer on a baking sheet, and roast for 20-30 minutes until lightly golden.

In a large saucepan, heat a bit more oil over medium heat. Sauté the diced onion for about 5-7 minutes until tender and translucent, then add the chopped garlic and thyme. Cook until fragrant, about 1 minute.

Add the roasted broccoli and broth to the saucepan. Bring this mix to a boil, then reduce the heat and let it simmer, covered, for 20 minutes.

Using a stick blender, carefully puree the soup until smooth and creamy.

Stir in the shredded cheddar until it melts into the soup. Mix in your choice of milk or cream and season with salt and pepper to taste.