Preheat the oven to 350 degrees F. Lightly coat a 2-quart oval baking dish with olive oil. Bring a large pot of water to a boil and add enough salt so that the water tastes like the sea. Add the noodles and cook 2 minutes less than the package directions. The noodles should be firm-tender. Reserve 1/2 cup pasta water, and then drain the noodles. Let the noodles and pasta water cool slightly. In a large bowl whisk the eggs with the broth and pasta water until smooth. Add the noodles and nutmeg and stir to combine. Set aside.
Meanwhile, heat 2 tablespoons olive oil in a large skillet over medium heat. Add the shallots and cook, without stirring, until the edges are browned, about 5 minutes. Stir the shallots and continue to cook, stirring, until well browned. Add the chard stems, garlic 1/2 teaspoons salt and pepper to taste. Cook until the stems are softened, about 5 minutes. Stir in the chard leaves and a splash of water and cook until wilted, 2 to 3 minutes. Remove from the heat, cool slightly and then stir in the dill.
Add the chard mixture to the noodles and season with 3/4 teaspoon salt and pepper to taste. Transfer to the prepared baking dish and drizzle with the remaining tablespoon olive oil. Place the baking dish in a large roasting pan and add enough water so it comes up the side of the pan about one-third of the way.
Carefully transfer to the oven and bake until the center is set, about 1 hour 20 minutes, tenting the kugel with foil when the noodles are golden brown and crisp. Let stand at least 5 minutes. Garnish with dill. Serve hot or at room temperature.