Begin by melting 1 1/2 tablespoons of butter in a large pot over medium heat. Add the finely chopped carrots, celery, and onion, sautéing them for about 3 to 4 minutes until they soften. Introduce the minced garlic and continue sautéing for an additional 30 seconds.

Now, stir in the chicken broth, cubed potatoes, dried thyme, and season the mixture with salt and freshly ground black pepper according to your taste. Allow the concoction to come to a gentle boil over medium-high heat. Once it's bubbling, reduce the heat to medium, cover the pot with a lid, and let it simmer for a delightful 15 minutes. Stir in the chopped broccoli and let it cook for an additional 5 minutes or until all the veggies turn tender.

Meanwhile, in a separate medium-sized saucepan, melt the remaining 4 tablespoons of butter over medium heat. Gradually whisk in the all-purpose flour and continue whisking constantly for approximately 1 minute. While whisking vigorously, pour in the milk, ensuring there are no lumps in the mixture.

Keep stirring the mixture continuously until it thickens. At this point, introduce the heavy cream to the mixture. Once the vegetables in the pot are tender, carefully pour the creamy mixture into the soup. Stir well to incorporate.

Remove the pot from heat and introduce the shredded sharp cheddar cheese and finely shredded parmesan cheese. Stir until the cheeses melt and meld perfectly into the soup. The result is a velvety, luscious blend of flavors.

Your Cheddar Broccoli Potato Soup is now ready to be savored. Ladle it into bowls, serve it warm, and relish every spoonful of this heartwarming concoction.