To prepare the soup, preheat the oven to 375°.
Remove white papery skin from garlic head (do not peel or separate cloves). Cut top third off head, and discard (A). Drizzle 1 teaspoon oil over cut side of garlic (B); wrap in foil. Bake at 375° for 50 minutes, cut side up, or until tender and cut side is lightly golden; cool 10 minutes. Separate cloves, and squeeze to extract garlic pulp (C). Discard skins.
Increase oven temperature to 400°.
Combine 2 teaspoons of oil, yellow onions, and next 4 ingredients (through pepper); toss gently. Arrange on a jelly-roll pan coated with cooking spray. Bake at 400° for 30 minutes or until tender and golden brown. Remove from oven, and cool slightly. Coarsely chop onions and shallots.
Heat 1 1/2 teaspoons oil in a large Dutch oven over medium heat. Add onions, shallots, and garlic; cook 10 minutes, stirring occasionally. Add beef broth and rice; bring to a boil. Cover, reduce heat, and simmer 15 minutes or until rice is tender. Add chicken broth, and cook 5 minutes or until thoroughly heated.
Place one-fourth broth mixture in a blender. Remove center piece of blender lid (to allow steam to escape); secure blender lid on blender. Place a clean dishtowel over opening in blender lid (to avoid spills). Process until smooth. Pour the pureed broth mixture into a large bowl. Repeat procedure in batches with the remaining broth mixture. Stir in half-and-half.
To prepare toasts, preheat broiler.
Sprinkle cheese evenly over bread slices. Place bread slices on a baking sheet. Broil 5 minutes or until cheese melts and is lightly browned. Ladle soup into bowls; sprinkle with chives. Serve toasts with soup.