Combine 2 cups stock and rice in a small saucepan over medium-high heat; bring to a boil. Cover, reduce heat, and simmer 25 minutes. Remove from heat; let stand 5 minutes. Place rice mixture and milk 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.
Heat a large saucepan over medium heat. Add oil to pan; swirl to coat. Add onion; sauté 4 minutes, stirring occasionally. Add garlic; cook 30 seconds. Add broccoli and salt; cook 5 minutes, stirring frequently. Add remaining 2 cups stock; bring to a boil. Reduce heat, and simmer 5 minutes or just until broccoli is tender. Add rice mixture; simmer 2 minutes, stirring occasionally.
Place 2 cups soup in blender; process until smooth. Return pureed soup to pan. Add pepper and 4 ounces cheese; stir until cheese melts. Divide soup evenly among 6 bowls; sprinkle evenly with remaining 1 ounce cheese.
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