Melt the butter in a large pot over low heat, then add the cauliflower, onion, garlic, and thyme and cover the pot for 5 minutes. The onion should start to soften and smell sweet and nutty, and the thyme and garlic should smell fragrant but not browned.

Pour in the chicken broth, raise the heat to bring the pot to a boil, then lower it to a simmer and cook for about 30 minutes, stirring now and then. The cauliflower is ready when a fork slips through the florets easily and they break apart with almost no resistance.

Take the pot off the heat and blend the soup until it is very smooth with an immersion blender, or purée it in batches in a blender. The finished base should look velvety and thick enough to coat the back of a spoon like heavy cream.

Set the pot over medium-low heat and whisk in the cream cheese and shredded cheeses until melted, about 5 minutes, then stir in the heavy cream and season with salt and pepper to taste. Keep the soup below a boil so the cheese melts into a glossy, creamy base instead of turning grainy or splitting.