Preheat the oven to 400˚F. Bring a large pot of water to a boil.
In the meantime, put 2 tablespoons of the coconut milk in a small bowl and add the cornstarch. Stir until dissolved. Set aside. Pour the rest of the coconut milk into a medium saucepan. Add the grated garlic and ginger. Bring to a boil, reduce to a simmer and cook 10 minutes, stirring occasionally.
Give the cornstarch coconut mixture another stir, then whisk it into the simmering coconut milk. Bring back to a simmer and cook, whisking, 2-4 minutes, or until thick and rich. Season with sea salt.
Once the water reaches a boil, season it generously with salt. Add the Swiss chard and cook until tender, 3-5 minutes. Drain and rinse with cold water. Using your hands, squeeze the water out of the chard. Finely chop the chard and add it to the coconut milk. Stir to combine. Squeeze in the juice from a quarter of a lime, and season to taste with salt and pepper. Transfer to a 1-quart gratin or baking dish. (See the do-ahead tip in the notes below.)
In a small bowl, combine the breadcrumbs and olive oil and toss with your fingers until evenly coated. Season with salt and pepper. Sprinkle the breadcrumbs evenly over the top of the gratin.
Bake 12-15 minutes, or until bubbling and light golden on top. Let cool 5 minutes. Serve warm.