Heat 1 tablespoon oil in a medium ovenproof nonstick skillet over medium-high heat. Add the celery, onion and green peppers and cook, stirring, until tender, about 5 minutes. Remove from the heat and transfer the vegetables to a plate and set aside. Wipe out the skillet.
Whisk the milk, eggs, 3/4 teaspoon salt and a few grinds of pepper in a large bowl. Add the cooked vegetables, shrimp, parsley and seafood seasoning and stir to combine.
Heat 1 tablespoon oil in the skillet over medium-high heat. Add the egg mixture and cook until the edges start to set, about 2 minutes. Transfer to the oven and bake until the center is set, 15 to 20 minutes. Let stand for 5 minutes before sliding carefully onto a serving plate and cutting into wedges.
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