Melt the unsalted butter in a large stockpot or Dutch oven over medium heat.

Add the diced sweet onion and cook until translucent, about 2 to 3 minutes.

Stir in the minced garlic and dried thyme, cooking for an additional minute.

Whisk in the all-purpose flour until lightly browned, around 1 minute.

Incorporate the chicken stock, scraping any browned bits from the bottom. Then add the diced potatoes, seasoning with salt and pepper to taste.

Bring the mixture to a boil, then reduce heat and let it simmer until potatoes are tender, about 20 minutes.

Mash some of the potatoes lightly to thicken the soup, if desired.

Stir in the chopped broccoli florets and heavy cream. Cover and cook until the broccoli is tender, about 8 to 10 minutes.

Remove from heat, let cool for 2 minutes, then gradually stir in the shredded cheddar cheese until melted.

Season to taste with additional salt and pepper before serving.