Heat 2 tablespoons of the oil in a skillet over medium heat until hot. Add the onions and reduce the heat to medium. Sprinkle with 1/2 teaspoon salt and a large pinch of pepper. Cook, stirring, until the onions are soft and beginning to brown, about 3 minutes. Add the celery and cook, stirring occasionally, until the celery is just beginning to soften, about 2 minutes. Transfer the mixture to a large bowl to cool to room temperature, about 5 minutes.
Lightly beat one of the eggs in a small bowl and add it to the bowl with the onion-celery mixture. Add the chicken, hot sauce, scallions and 3/4 teaspoon salt and stir until evenly combined.
Lightly beat the remaining egg with 1 tablespoon water in a small bowl. Working with 1 dumpling wrapper at a time on a clean surface (cover the remaining wrappers with a damp paper towel so they don't dry out), add 1 rounded teaspoon of the filling to the center of the wrapper. Brush the edge of the wrapper with the egg wash. Fold the dumpling into a half-moon shape and press the edges to seal. (Alternatively, you can seal the edge tightly by forming pleats.) Transfer the dumpling to a baking sheet and cover with a damp paper towel to keep from drying out while you finish assembling the remaining dumplings.
Heat 2 tablespoons of the oil in a large skillet over medium-high heat. Working in 3 batches and being careful not to crowd the skillet, add a third of the dumplings in a single layer. Sear each side until crisp and brown, about 30 seconds per side. Reduce the heat to low, add 2 tablespoons water to the skillet and cover with a lid to steam. Cook until the dumplings are soft and cooked through and the water is completely evaporated, 3 to 5 minutes. Transfer to a serving platter and repeat with the remaining dumplings, adding additional oil and water to the skillet for each batch.
Serve with celery and carrot sticks and blue cheese dressing for dipping.
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