Preheat the oven to 400 degrees F. Butter a 3-quart baking dish.
Remove the stems from the chard leaves and chop the stems. Heat the oil in a large skillet over medium heat and saute the stems, shallot, 1 teaspoon salt and a couple turns of black pepper until the shallot is golden, about 8 minutes. Stir in the chard leaves and cook, turning with tongs, until the leaves are wilted, about 4 minutes.
Transfer the chard to the prepared baking dish and make 6 shallow wells in the leaves. Crack an egg into each well and sprinkle the eggs with salt and black pepper. Bake, rotating the dish as needed for even cooking, until the egg whites are set but the yolks are still runny, 10 to 12 minutes.
Stir together the yogurt and lime juice in a small bowl; season with salt and black pepper. Top the eggs with dollops of the yogurt and sprinkle with the red pepper flakes. Serve with bread on the side.