In a large Dutch oven, heat the olive oil over medium heat. Add the chopped onion, carrots, and fresh thyme. Stir to coat, then let cook about 10 minutes, stirring occasionally, until onions are a little brown and starting to caramelize. You're trying to get good flavor here, so don't stir too often; let them brown.
Add the cauliflower florets to the Dutch oven in batches of 1-2 cups at a time.** Stir to coat then let brown a little bit, stirring only occasionally, before adding the next batch of florets. The onions will continue to brown, and that's okay. Adjust heat as needed so veggies brown, not burn.
After all the cauliflower has been added and has browned, add a splash of the chicken broth to deglaze the pan, scraping the bottom of the pot with a wooden spoon or spatula. You want all that flavor from the bottom of the pot going into the soup.
Add the chopped potatoes to the pot, then add the remaining chicken broth.
Add the salt and pepper to the pot, stirring carefully to ensure that all veggies are mostly covered by the broth.
Bring mixture to a boil, then reduce heat to low and cover. Simmer 25-30 minutes, or until cauliflower and potatoes are tender when pierced with a fork.
Remove the soup from the heat and turn off the stove. Let cool 10-15 miuntes.
Use a stick blender/immersion blender to blend the soup until thick and completely smooth.
Sprinkle the 3 cups grated cheese into the hot soup and stir until completely melted. Taste, then add additional salt and/or pepper, if needed.
Ladle a generous portion of soup into a bowl, then serve hot. Garnish with chopped chives, extra grated cheddar cheese, or a sprinkling of paprika.