Heat avocado oil in a large Dutch oven over medium/high heat. When the oil is fragrant, add the onion, celery, and carrots to the pot and cook for 2-3 minutes.
Next, add the butter, garlic, salt, pepper, and thyme to the Dutch oven. Cook for 1-2 minutes, allowing the garlic to become fragrant and golden brown.
Add the potatoes and chicken broth. Use a wooden spoon to scrape up any brown bits that may have developed on the bottom of the Dutch oven. Bring the broth to a boil, then cover and reduce heat to low, and let simmer for 20 minutes.
Add the broccoli to the soup and continue to cook uncovered on low for 10 minutes.
Remove the Dutch oven from the heat. Using an immersion blender give the soup a couple of pulses to break up the potatoes and broccoli leaving it chunky.
Put the Dutch oven back on the stove and turn the heat to low. Add in the heavy cream and shredded cheddar cheese. Heat until the cheese is melted, 4-5 minutes.