1-First Step: Start by gathering all your ingredients. Having everything measured and chopped before you turn on the stove, also known as mise en place, makes the cooking process much smoother. Chop your onion, mince your garlic, cut your broccoli into florets, and julienne your carrot. Grate your cheddar cheese and set it aside so it is ready to melt when needed.

2-Second Step: Melt the butter in a large pot or Dutch oven over medium heat. Once the butter is bubbling, add the chopped onion, salt, and a generous pinch of freshly ground black pepper. Cook this mixture for about 5 minutes, stirring occasionally, until the onion has softened and become translucent. This creates a flavorful base for the soup.

3-Third Step: Stir in the chopped garlic and cook for just 1 minute until fragrant. Be careful not to burn the garlic, as it can turn bitter. Next, sprinkle the flour over the onion and garlic mixture. Whisk continuously for 1 to 2 minutes. The flour will bubble and turn a golden color, cooking out the raw flour taste and forming a roux that will thicken our soup.

4-Fourth Step: Slowly pour in the milk while whisking continuously. It is important to pour slowly and whisk constantly to prevent lumps from forming. Once the milk is incorporated and smooth, add the vegetable broth. Now, toss in the broccoli florets, julienned carrot, and the Dijon mustard. Stir everything together to combine.

5-Fifth Step: Bring the soup to a gentle simmer, then reduce the heat to medium-low. Let it simmer uncovered for 15 to 20 minutes. You want the broccoli to become tender but not mushy. The carrots should be soft as well. As the soup simmers, the flavors will meld together, and the liquid will reduce slightly, making it richer.

6-Final Step: Turn the heat down to very low. Gradually add the shredded cheddar cheese, a handful at a time, stirring between each addition until melted and creamy. Do not rush this step; adding the cheese too quickly or over high heat can cause the sauce to separate or become oily. Once all the cheese is melted and smooth, taste the soup and season with more salt and pepper if needed. Serve hot, topped with croutons if you like, and enjoy!