In a small saucepan, melt 1 tablespoon butter over medium heat. Add the diced onion and sauté until translucent and barely browned, about 4 minutes. Stir intermittently.

Add the garlic and cook for about 30 seconds, stirring constantly. Remove from heat and set aside.

In a large heavy-bottomed pot, melt 4 tablespoons butter, add flour, and cook over medium heat, whisking constantly for 3 to 5 minutes until thickened. This creates a roux.

Gradually whisk in the vegetable stock, then the half-and-half, ensuring smooth incorporation.

Simmer the mixture over low heat for 15 to 20 minutes, stirring occasionally. The mixture should reduce and thicken slightly.

While simmering, chop the broccoli and carrots. Add the broccoli, stems, onion, and garlic to the pot. Stir to combine.

Add salt, pepper, paprika, dry mustard, and cayenne if using. Stir well.

Continue to simmer for another 20 to 25 minutes, stirring occasionally, until the vegetables are tender and the soup is slightly thickened.

Grate most of the cheese and stir into the soup until melted. Reserve some cheese for garnish.

Ladle the soup into bowls, garnish with reserved cheese, and serve immediately.