Heat oven to 375° F.
In a medium bowl, combine the oil, vinegar, tomatoes, and 1 tablespoon of the chives.
Arrange the bread in a single layer in a 13-by-9-inch ovenproof baking dish. Top with the prosciutto and the 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.
Sprinkle with the remaining tablespoon of chives.