Melt one to two tablespoons of butter in a skillet over medium-high heat.
Sauté the diced onion in the melted butter until it begins to soften, which should take about 3 minutes.
Add the sliced carrots, salt, and pepper, and continue to cook for another 3-4 minutes.
Then, add the minced garlic and sauté while stirring for 30 seconds.
Next, incorporate the diced potatoes and pour in the chicken broth.
Cover the skillet and bring the mixture to a simmer. Allow it to simmer for about ten minutes.
Add the broccoli florets and continue to simmer until both the broccoli and potatoes are tender, which will take about 10 minutes. The potatoes should be fork-tender.
In a small bowl, mix the cornstarch into the milk until smooth. Then, stir this mixture into the hot soup.
Finally, add the shredded cheddar cheese and stir until it is fully melted.
Serve the soup hot and enjoy!