Preheat oven to 450°.
Arrange bread cubes in a single layer on a jelly-roll pan; coat bread cubes with cooking spray. Bake at 450° for 10 minutes or until toasted. Set aside.
Heat a Dutch oven over medium-high heat. Coat pan with cooking spray. Add onion; sauté 4 minutes. Add garlic; sauté 1 minute. Stir in beer; bring to a boil. Reduce heat; simmer 20 minutes or until onion is very tender.
Place beer mixture and 1 cup broth in a blender. Remove center piece of blender lid (to allow steam to escape); secure blender lid on blender. Place a clean towel over opening in blender lid (to avoid splatters). Blend until smooth. Return pureed mixture to pan. Stir in remaining 3 cups broth; bring to a boil. Reduce heat, and simmer 10 minutes.
Lightly spoon flour into a dry measuring cup; level with a knife. Combine flour and 1 cup milk, stirring with a whisk until smooth. Add flour mixture and remaining 1 cup milk to pan; cook 12 minutes or until slightly thickened. Remove pan from heat. Gradually add cheese, stirring until smooth. Stir in pepper. Serve with bread cubes and chives.