Begin with Butter: In a large pot or Dutch oven, heat the unsalted butter over medium heat. Once melted, introduce the chopped yellow onion, sprinkle with sea salt, and add a few twists of freshly ground black pepper. Cook the onions until they're soft and translucent, stirring occasionally, which should take about 5 minutes.
Add Aroma with Garlic: Follow by adding the chopped garlic cloves to the onions. Let them cook together for about a minute, just until the garlic releases its fragrance.
Flour for Thickness: Sprinkle the all-purpose flour evenly over the onion and garlic mixture. Keep whisking for a couple of minutes until the flour turns a golden brown color, ensuring a roux is formed without any lumps.
Milk Mixture: Slowly add the whole milk or unsweetened almond milk to the pot. It's important to continue whisking while pouring the milk to integrate it smoothly into the roux and avoid any clumps.
Vegetable Simmer: Introduce the vegetable broth into the mixture, followed by the broccoli florets, finely chopped or julienned carrot, and a hint of Dijon mustard. Stir everything together, making sure the ingredients are well combined. Allow the soup to simmer gently for 15 to 20 minutes. The key is to cook until the broccoli becomes perfectly tender.
Cheese to Creaminess: To finish, gradually incorporate the shredded cheddar cheese into the soup. Stir steadily after each addition until the cheese is completely melted and the soup achieves a creamy consistency. Taste and adjust the seasoning as needed. Serve the soup warm, optionally topped with homemade croutons for an extra touch of texture and flavor.