How to Make It
In a medium saucepan, melt 4 tablespoons of the butter. Add the shallot and cook over moderate heat, stirring, until softened, about 2 minutes. Add the garlic, tomatoes, tomato paste and wine and bring to a boil. Simmer over moderately high heat, stirring, until nearly all the wine is absorbed and the tomatoes just start to pop, 4 to 5 minutes. Add the oats and chicken stock and cook over moderate heat, stirring occasionally, until the oats are just tender and suspended in a creamy sauce, 8 to 10 minutes. Stir in the 1/4 cup of basil, 1/4 cup of grated cheese and the remaining 1 tablespoon of butter. Season the oatmeal with salt and pepper and keep warm, adding tablespoons of water if it becomes too thick.
Meanwhile, bring a large, deep skillet of water to a simmer over moderate heat. Add the white vinegar and a generous pinch of salt. Crack the eggs into a small bowl, one at a time, and carefully slide them into the simmering water. Poach the eggs over moderate heat until the whites are set but the yolks are still slightly runny, about 4 minutes. Transfer the poached eggs to paper towels to drain.
Spoon the oatmeal into shallow bowls. Top with the poached eggs, garnish with basil and serve right away, passing additional cheese at the table.