Whisk the egg white with 1 tablespoon each rice wine, soy sauce and cornstarch in a large bowl. Add the shrimp and toss; refrigerate until ready to fry. Whisk the chili sauce, the remaining 2 tablespoons each rice wine and soy sauce, 1 tablespoon cornstarch and 1 cup water in a bowl; set aside.
Heat 2 tablespoons oil in a wok or large Dutch oven over medium-high heat. Add the broccolini; stir-fry until charred in spots, 3 minutes. Add 1/2 cup water and stir-fry until crisp-tender, 1 to 2 minutes; transfer to a plate. Reduce the heat to medium; add the remaining 1 tablespoon oil, the scallions, garlic and red pepper flakes. Stir-fry until tender, 2 minutes. Add the chili sauce mixture and cook, stirring occasionally, until thickened, 1 minute. Cover and set aside.
Heat 1 inch of oil in a deep saucepan over medium-high heat until a deep-fry thermometer registers 365 degrees F. Working in batches, toss the shrimp in cornstarch to coat, then fry until crisp and golden, about 3 minutes. Remove with a slotted spoon and drain on paper towels.
Return the wok to medium heat and reheat the sauce if needed. Add the broccolini and shrimp and cook, tossing, until warmed through, 2 minutes. (Add up to 1/2 cup water if the sauce is too thick.) Serve with the rice.