1. 1. Saute Aromatics: In a large pot or Dutch oven, melt the unsalted butter over medium heat. Add the chopped yellow onion and cook until softened and translucent, about 5-7 minutes. Stir in the minced garlic and cook for another minute until fragrant, being careful not to burn it. The onions should look soft and slightly shiny.

2. 2. Add Broth and Broccoli: Pour in the chicken or vegetable broth and bring it to a gentle simmer. Add the chopped fresh broccoli florets. Cover the pot and cook for 8-10 minutes, or until the broccoli is tender when pierced with a fork. It should be bright green and easily mashable.

3. 3. Create Creamy Base: While the broccoli cooks, combine the half-and-half (or whole milk) and cottage cheese in a blender. Blend until completely smooth and no lumps of cottage cheese remain. This step is crucial for achieving a silky, uniform texture in your soup.

4. 4. Combine and Season: Once the broccoli is tender, remove the pot from the heat. Stir in the blended half-and-half and cottage cheese mixture, Dijon mustard, salt, and black pepper. Mix well to combine all ingredients thoroughly.

5. 5. Melt the Cheese: Gradually add the shredded cheddar cheese to the pot, stirring constantly until it is fully melted and incorporated into the soup. Do this off the heat or over very low heat to prevent the cheese from clumping or separating. The soup should become thick and smooth.

6. 6. Adjust and Serve: Taste the soup and adjust the seasonings if needed. If you prefer a smoother soup, you can use an immersion blender to blend some or all of the broccoli directly in the pot. Serve hot and enjoy!