Heat oven to 375º F. In a medium bowl, combine the oil, vinegar, tomatoes, and 1 tablespoon of chives.
Arrange the bread in a single layer in a 13-by-9-inch ovenproof baking dish. Top with the prosciutto and tomato mixture.
Break the eggs on top and drizzle with the cream. Bake until the whites have set but the yolks are slightly runny, 15 to 20 minutes. Garnish with the remaining chives.