In a food-processor bowl or blender container, combine the 1/2 cup carrots and the onion. Cover and process until nearly smooth (add small amount of the chicken stock to thin the mixture, if needed). Set aside.
In a large saucepan, melt butter. Stir in flour until smooth. Cook over medium-low heat, stirring occasionally, for 10 minutes or until mixture turns a copper color.
Carefully stir in chicken stock or broth, cream, beer, carrot-onion mixture, pepper, pepper sauce, and bay leaf. Bring to boiling; reduce heat. Simmer, uncovered, for 10 minutes.
Stir the cornstarch and water together. Add to the soup. Cook and stir until mixture is thickened and bubbly. Discard bay leaf. Gradually add cheddar and Stilton cheeses, cooking and stirring until they're melted. Serve in warm bowls, topped with chopped carrot and chives. Makes 4 to 6 side-dish servings.