The secret ingredient here is Dijon mustard. You won’t taste it, but it’s working behind the scenes to deepen the flavor. Chop the chard with your longest chef’s knife for efficiency—a 10-incher is perfect.

Preheat the oven to 350°F.

Trim the stems from the chard leaves. Discard the stems. Immerse the leaves in water to rid them of grit. Lift them out and drain. Chop the leaves coarsely.

Heat 3 tablespoons of the oil in a 10-inch ovenproof skillet over medium-high heat. Add the mushrooms, sprinkle lightly with salt, and sauté, stirring frequently, until golden brown, 5 to 7 minutes. Lower the heat to medium, add the shallots, and cook until the shallots are softened but not browned, about 30 seconds. Add the chard a few handfuls at a time and cook, tossing with tongs, until soft and wilted, about 4 minutes. Add ¼ teaspoon of the salt and continue cooking and tossing until all the liquid in the pan evaporates, about 3 more minutes. Turn off the heat.

Whisk the eggs, half-and-half, mustard, the remaining ½ teaspoon salt, and several grinds of pepper together in a medium bowl. Stir in the cheese. Add the remaining 1 tablespoon olive oil to the skillet and heat over medium heat. Spread out the vegetables evenly and, when the pan is hot, pour in the egg mixture. Cook until the bottom is set, about 3 minutes, and then transfer the skillet to the oven. Bake until the eggs are set on top, about 15 minutes.

Place the frittata under the broiler a few inches from the heat source until the top is golden, 2 to 3 minutes. Remove from the heat and let rest for a few minutes; the frittata will pull away from the sides of the pan. Slice in the pan or else flip the frittata onto a plate and serve.