Adjust oven rack to upper-middle position and preheat oven to 375 F degrees.
Stem chard, cut stems into 2-inch pieces and set aside. Slice leaves into 1-inch-wide strips.
Steam the greens: Bring 2 cups of water to a boil in a large pot (I used a 7 qt Staub) over high heat. Add kale, cover, and reduce heat to medium-high. Cook until kale is wilted, about 5 minutes, stirring halfway through cooking. Add chard leaves, cover, and continue to cook until chard is wilted, about 4 minutes longer, stirring halfway through cooking. Drain greens in a colander and let it sit in the sink while you proceed with the rest of the recipe. Do not wash the pot.
Pulse garlic in a food processor until coarsely chopped. Add bread and 3 tablespoons oil and pulse a couple times until the largest crumbs are smaller than ¼ inch (most will be much smaller). Add Parmesan, ¼ teaspoon salt, and ¼ teaspoon pepper and pulse to combine. Transfer to a medium bowl. Do not wash out the food processor.
Add onion and chard stems to the food processor and process until finely chopped, scraping down the sides of the bowl as needed. Warm remaining 2 tablespoons oil over medium heat in the pot you used earlier to cook the greens. Add onion, chard stems, thyme, remaining ½ teaspoon salt, and remaining ¼ teaspoon pepper. Cook over medium-high heat, stirring occasionally, until onion and chard stems are tender, 8 to 10 minutes.
Using a spatula, press gently on greens in colander to remove excess moisture. Transfer greens to pot with onion mixture. Remove from heat. Add cream, nutmeg and cayenne and stir to combine. Taste and adjust seasoning accordingly. Transfer mixture to a gratin dish or something similar to an 11 x 7. Sprinkle the top with gruyere. Spread bread-crumb mixture evenly over the filling. Bake in the upper-middle of the oven until topping is golden brown and filling bubbles around edges, 20 to 25 minutes.