In a Dutch oven set over medium heat, cook the bacon until crisp, 8 to 10 minutes. Transfer the bacon to a paper towel-lined plate, set aside, and pour off all but 1 tablespoon of bacon drippings from the pan.

Add the chopped onion, celery, and garlic to the pan and drippings and cook, stirring frequently, until just beginning to become tender, about 5 minutes.

Add the chopped cauliflower, the chicken broth, salt, pepper, and fresh thyme. Bring the broth to a boil; cover and reduce the heat to medium-low and simmer until the cauliflower is completely tender, about 15 minutes.

Let the mixture cool slightly then carefully pour it into a blender. Add the whole milk. Attach the blender lid on top, removing the center piece of the lid to let steam escape. Cover that center opening with a towel, holding your hand over the towel, and process the soup mixture on low at first (gradually increasing the speed) until completely smooth, 1 to 2 minutes. Pour the smooth soup back into the Dutch oven on the stovetop and set it over medium heat to let it warm through, about 5 minutes. Taste and adjust seasonings, adding more salt and pepper, if needed

Crumble the cooked bacon slices. Serve the soup in bowls topped with crumbled bacon and shredded cheddar cheese.