Dice the onion, peel and cube the potatoes into ½-inch pieces, dice the carrots, and mince the garlic. If using fresh broccoli, cut into bite-sized florets. Having everything prepped makes the cooking flow smoothly.

In a large pot or Dutch oven, melt 1–2 tablespoons of butter over medium-high heat. Once shimmering, add the diced onion.

Sauté the onion, stirring occasionally, until it begins to soften and turn translucent — about 3 minutes.

Stir in the diced carrots with ½ teaspoon salt and ¼ teaspoon pepper. Continue to cook for 3–4 minutes until the carrots begin to soften.

Add the minced garlic and sauté, stirring, for about 30 seconds until fragrant.

Pour in the cubed potatoes and the 4 cups of chicken stock. Bring the pot to a coverable simmer, then reduce the heat and cover. Let the soup simmer for about 10 minutes.

Add the broccoli florets and continue simmering with the lid off or slightly ajar until both the potatoes and broccoli are fork-tender — about another 8–10 minutes.

While the vegetables finish cooking, whisk ¼ cup cornstarch into 1 cup of cold milk in a small bowl until completely smooth.

Slowly pour the cornstarch-milk mixture into the hot soup, stirring constantly. Increase the heat to medium and simmer gently for 2–3 minutes until the soup thickens and becomes glossy.

Lower the heat to medium-low and add 1½ cups shredded sharp cheddar cheese in handfuls, stirring until each addition melts fully before adding the next.

Taste the soup and adjust seasoning with more salt or pepper if needed. Serve warm and garnish as desired.