Heat a large Dutch oven over medium heat. Add oil to pan; swirl to coat. Add onion; sauté 10 minutes or until browned. Stir in red pepper and garlic; cook 1 minute. Add broth and 1 cup water; bring to a boil. Stir in bulgur and salt. Cover and simmer 10 minutes. Add broccoli rabe. Cover and simmer 12 minutes or until broccoli rabe is tender; stir in lemon juice. Ladle stew evenly into 6 shallow bowls.
Add water to a large skillet, filling two-thirds full; bring to a boil. Reduce heat; simmer. Add vinegar. Break each egg into a custard cup, and pour each gently into pan; cook 3 minutes or until desired degree of doneness. Carefully remove eggs from pan using a slotted spoon; place 1 poached egg on each serving of stew. Sprinkle evenly with cheese and black pepper.
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