Cut lemon lengthwise into 4 wedges; remove and discard seeds. Set aside 1 lemon wedge for serving; reserve remaining wedges for another use.

Place a 10-inch cast-iron skillet in oven. Preheat oven to 425°F. (Do not remove skillet while oven preheats.)

Whisk together flour, ¾ teaspoon of the kosher salt, and ½ teaspoon of the pepper in a large bowl. Whisk together eggs, milk, and mustard in a medium bowl until completely combined. Gradually whisk about half of the milk mixture into flour mixture, whisking until smooth and free of lumps. Gradually whisk in remaining milk mixture until thoroughly combined.

Carefully remove hot skillet from preheated oven. Add 3 tablespoons of the butter to skillet, and swirl to coat. Working quickly, pour batter into skillet. Sprinkle with cheese, leaving a 1 ½-inch border around edges. Bake in preheated oven until puffed and brown around edges, 20 to 25 minutes.

Meanwhile, heat oil in 12-inch skillet, preferably cast-iron, over high until oil just begins to smoke. Add mushrooms, and stir to coat in oil. Cook, undisturbed, until bottom side is browned, 2 to 3 minutes. Stir mixture, and sprinkle with remaining ¾ teaspoon kosher salt and ¼ teaspoon pepper. Cook, stirring occasionally, until mushrooms are tender and browned all over, about 4 minutes. Reduce heat to medium. Add shallot and remaining 2 tablespoons butter; cook, stirring constantly, until butter is melted, about 30 seconds. Remove from heat; let cool 3 minutes. Stir in arugula.

Remove Dutch baby from oven. Spoon mushroom mixture into crater in center. Squeeze reserved lemon wedge over mushroom mixture; garnish with flaky sea salt, and serve immediately.