To prepare marinade: Whisk vinegar, soy sauce, sesame oil, canola oil, brown sugar, and ginger in a medium bowl. To prepare bibimbap: Place mushroom caps in a shallow dish. Pour in about half the marinade and turn to coat. Set aside. Add shrimp to the remaining marinade; coat evenly. Cover and refrigerate for at least 30 minutes or up to about 1 hour, while you prepare the rice and vegetables. Bring water to a boil in a large saucepan over medium heat. Stir in rice, reduce heat to low, cover, and simmer at the lowest bubble until the rice is tender, about 50 minutes. Let stand, covered, until ready to serve. Meanwhile, cut carrot, cucumber, and radishes into 1- to 2-inch matchsticks (or cut into julienne strips using a mandoline slicer); set aside in separate piles along with the chopped bean sprouts. About 20 minutes before you're ready to grill, preheat a gas grill to high or prepare a charcoal fire. If grilling on a gas grill, keep one half on high heat and turn the other half down to medium heat. If using charcoal, move the coals into two separate piles. One pile should be larger for high heat and the other smaller for medium heat. Brush the grill rack clean and oil it (see Tip). Grill the mushroom caps directly over medium heat until soft and tender, basting occasionally with the marinade in the dish and turning once or twice, 8 to 12 minutes. Grill the shrimp directly over high heat until lightly charred and opaque in the center, turning once, 2 to 3 minutes. Remove to a clean cutting board. Cut the mushrooms and shrimp into 1/2-inch pieces; keep separate. Get each portion ready before cooking eggs. Put 1 cup of the warm rice in each of 4 bowls. Arrange the vegetables, mushrooms, and shrimp in separate piles around the edge. Heat 1 tablespoon oil in a large nonstick skillet over medium heat. Crack eggs into the pan, being careful not to break the yolks. Cook until the whites are set, 3 to 5 minutes, or to desired doneness. Place a fried egg in the center of each portion. Each diner should stir everything in the bowl together, adding chile paste to taste. Exchanges: 3 starch, 1 vegetable, 3 1/2 lean meat, 3 fat. Carbohydrate Servings: 4. Nutrition bonus: Vitamin A (61% daily value), Magnesium (38% dv), Iron (31% dv), Potassium and Zinc (24% dv), Folate (18% dv), Vitamin C (15% dv).
