Heat the olive oil in your soup pot over medium heat, then add the chopped onion. Cook it for about 5 minutes, stirring occasionally, until it turns soft and translucent. Stir in the minced garlic during the last 30 seconds so it doesn't burn and turn bitter.
Stir the tomato paste right into the softened onions and garlic, letting it cook for a minute to deepen its flavor. Pour in the fire-roasted tomatoes, breaking them up slightly with your spoon as they heat through in the pot.
Add the chicken or vegetable stock, dried thyme, salt, and pepper, then stir everything together well. Bring the mixture to a gentle boil, then lower the heat and let it simmer uncovered for about 12 to 15 minutes.
Remove the pot from the heat and let it cool for a few minutes before blending. Use an immersion blender directly in the pot, or carefully transfer the soup in batches to a countertop blender, until it's completely smooth.
Return the blended soup to low heat and stir in the freshly shredded cheddar a handful at a time, letting each addition melt fully before adding more. Finish by stirring in the milk or cream, then taste and adjust the salt and pepper before ladling it into bowls.