Melt the unsalted butter in your large Dutch oven over medium heat. Add the diced yellow onion and minced garlic, stirring constantly for four minutes until they become soft, translucent, and highly fragrant.

Pour the chicken broth into the pot with the sautéed onions and garlic. Add the cubed Russet potatoes, dried thyme, smoked paprika, salt, and black pepper, then bring the liquid to a boil.

Lower the stove heat to medium-low and cover the pot tightly with a lid. Let the potatoes simmer gently for twelve minutes until they are easily pierced with a fork.

Use a potato masher directly inside the pot to crush about half of the cooked potatoes. This technique creates a naturally thick and creamy base while leaving some chunky pieces for texture.

Stir the chopped fresh broccoli florets directly into the bubbling potato soup base. Cover the pot again and let it cook for five additional minutes until the broccoli turns bright green and tender.

Turn the stove heat down to the lowest setting completely. Pour in the whole milk or heavy cream, stirring the soup gently for one minute until the liquid is fully incorporated and warm.

Gradually add the freshly shredded sharp cheddar cheese one handful at a time. Stir continuously in one direction until the cheese melts completely into a perfectly smooth, velvet-like consistency.

Ladle the hot soup into deep serving bowls immediately while it is fresh. Garnish each portion with a tiny sprinkle of extra cheese and a few leftover broccoli bits if desired.