Season chicken cubes with salt and black pepper.
Mix cornstarch and water in a bowl until smooth; whisk in lemon juice, rice vinegar, ketchup, soy sauce, brown sugar, and chile pepper sauce into cornstarch mixture, stirring until brown sugar has dissolved and cornstarch is free of lumps.
Place a heavy skillet over medium-high heat; pour in vegetable oil. Add slices of red chili and ginger to the hot oil. Cook, stirring often, until oil is flavored, about 2 minutes. Stir in cashews and garlic; cook until cashews are lightly golden brown, about 2 minutes more.
Lightly toss seasoned chicken cubes with cashew mixture in skillet until combined, about 30 seconds. Turn heat down to medium-low and pour cornstarch mixture into skillet. Stir constantly until sauce has thickened, about 30 seconds. If too thick, add a splash of water. Simmer 2 more minutes to heat the chicken through. Adjust levels of salt, black pepper, soy sauce, and chile pepper sauce to taste. Stir in cilantro just until wilted, about 15 seconds.