In a small saucepan, add ¼ stick of butter, the diced onion, and sauté over medium heat until the onion is translucent and barely browned, about 5 minutes. Stir intermittently.

Add the garlic and cook about 60 seconds, stirring constantly so it doesn’t burn. Remove from heat and set pan aside.

In a large heavy-bottom pot, add full stick of butter, flour, and cook over medium heat for about 10 minutes, whisking constantly, until flour is thickened.

Slowly add the vegetable stock, whisking constantly.

Slowly add the cream, whisking constantly.

Allow mixture to simmer over low heat for about 15 to 20 minutes, or until it has reduced and thickened some. Whisk intermittently to re-incorporate the ‘skin’ that inevitably forms.

While mixture is simmering, chop the broccoli and carrots. After simmering 15 to 20 minutes, add the broccoli, carrots, and the onion and garlic you previously set aside.

Add the salt, pepper, optional paprika, optional dry mustard powder, and optional cayenne. Stir to combine.

Allow soup to simmer over low heat for about 20 to 25 minutes, or until it has reduced and thickened some. Whisk intermittently to re-incorporate the ‘skin’ that inevitably forms.

While soup simmers, grate the cheese. After simmering about 20 to 25 minutes, add most of the cheese, reserving a small amount for garnishing bowls. Stir in the cheese until melted and incorporated fully, less than 1 minute.

Transfer soup to bowls, garnish with reserved cheese, and serve immediately, preferably with bread.