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Preheat oven to 400°F. Bring a large pot of salted water to a boil over high. Cook egg noodles, stirring occasionally, until al dente, about 5 minutes; drain and rinse with cold water. Set aside.
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Combine mayonnaise, lemon zest and juice, 1/4 cup water, capers, black pepper, garlic, anchovies, and 2 cans tuna in a blender; process until smooth, about 25 seconds. Set tonnato sauce aside.
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Melt butter in a large Dutch oven over medium. Add onion; cook, stirring often, until softened, about 4 minutes. Sprinkle with flour; cook, stirring constantly, 1 minute. Stir in half-and-half; cook, stirring constantly, until thickened, 30 seconds to 1 minute. Stir in tonnato sauce, stock, and 3/4 cup Parmesan. Remove from heat; stir in egg noodles, peas, pimientos, salt, 2 tablespoons dill, and remaining 2 cans tuna. Transfer mixture to a 3-quart baking dish (such as a 9- x 13-inch dish).
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Top casserole evenly with potato chips and remaining 1/4 cup Parmesan. Cover dish tightly with aluminum foil, and bake in preheated oven for 15 minutes. Uncover, and continue baking until heated through, about 10 minutes. Garnish with remaining 1 tablespoon dill.