Preheat oven to 375 degrees. Melt butter in a medium skillet over medium-low heat. Add shallot. Cook, stirring constantly, until shallot has softened, about 2 minutes. Add mushrooms and a pinch of salt. Cook, stirring occasionally, until mushrooms have softened, about 5 minutes. Add cream; bring to a simmer. Immediately remove from heat, and let mushroom mixture cool 5 minutes.
Butter four 6-inch shallow baking dishes (such as gratin dishes). Place dishes on a rimmed baking sheet. Crack 1 egg into each baking dish, keeping yolks intact, and season with salt and pepper. Pour the mushroom-cream mixture into dishes, dividing evenly, and sprinkle with chives. Bake on baking sheet until the egg whites are just set, 9 to 12 minutes. Serve with bread, for dipping.