How to Make It
Preheat the oven to 325°. Spread the bread on a baking sheet and toast for about 10 minutes, until golden. Drizzle with 1 tablespoon of the olive oil and toss to coat.
Heat the remaining 2 tablespoons olive oil in a large nonreactive saucepan. Add the onion and garlic and cook over moderate heat, stirring, until golden, about 8 minutes. Add half of the tomatoes, the white wine and a large pinch of crushed red pepper and simmer until the liquid is reduced by half, about 15 minutes.
Add the chicken stock and bay leaves to the pan and bring to a boil over moderately high heat. Lower the heat and simmer for 15 minutes. Strain the broth into another saucepan, pressing down on the tomatoes. Season with salt and black pepper and keep warm.
In a large skillet, bring 6 cups of water and the white vinegar to a simmer. Crack the eggs into the water and poach until just set, about 3 minutes. Using a slotted spoon, transfer to a shallow bowl of ice water to cool. Place on baking sheets lined with paper towels.
To serve, divide the toasted bread, the uncooked tomatoes, the basil and half of the shaved Asiago among 6 bowls and add a poached egg to each. Pour the hot broth over and garnish with the remaining Asiago, a drizzle of extra-virgin olive oil and a pinch of crushed red pepper.