Melt the butter in a large pot over medium heat. Add the diced onion and cook until softened and translucent.
Add the carrots, salt, and pepper. Cook for several minutes until the carrots begin to soften.
Stir in the garlic and cook briefly until fragrant.
Pour in the chicken stock and add the potatoes. Bring to a gentle simmer, cover, and cook until the potatoes are fork-tender.
Add the broccoli and continue simmering until the broccoli is tender but still bright green.
Whisk the cornstarch and milk together until smooth, then stir into the soup to thicken.
Lower the heat and gradually stir in the shredded cheddar cheese until fully melted and smooth.
Taste and adjust seasoning as needed, then serve warm.
