Melt the butter in a large pot over medium heat. Add the finely chopped onion and cook until it becomes translucent. Next, stir in the minced garlic and sauté for another minute until fragrant.

Add the peeled and diced potato, chicken broth, and broccoli florets to the pot with the sautéed aromatics. Bring the mixture to a simmer and cook until the vegetables begin to soften. Halfway through the cooking time, add the diced carrots and continue to simmer until all the vegetables are completely tender, about 20 minutes total. I like to grate the carrots for a smoother texture in the final soup.

In a separate bowl, whisk together the evaporated milk and all-purpose flour until the mixture is completely smooth and free of lumps.

Pour the evaporated milk and flour mixture (from Step 3) into the pot with the simmered vegetables. Stir well and let the soup continue to cook until it thickens slightly and becomes creamy.

Reduce the heat to low and add the freshly shredded cheddar cheese and Dijon mustard. Stir until the cheese is completely melted and the soup is smooth and flavorful. If desired, season to taste with salt and pepper. Personally, I always taste at this stage and adjust with a little extra cheese or mustard if I feel the soup needs a flavor boost.

Ladle the hot broccoli cheddar soup into bowls and serve immediately, optionally with warm bread on the side. Enjoy!