Blend the eggs, milk, nutmeg, salt, pepper, and pureed onions in a wide bowl. Soak the bread rounds until they're wet with the custard mixture all the way through, but not falling apart.
Heat about a quarter-inch of the oil in a skillet until it shimmers at the surface. Fry the bread rounds until they are quite a dark brown on both sides, turning once. Drain on a paper towel and keep warm.
Spread a little sour cream on each bread round. Sprinkle with chives, and top with a generous spoonful of caviar.