Prepare the sauce: combine 3/4 cup water, pineapple juice, vinegar, soy sauce, ketchup, both sugars, and salt in a medium saucepan; keep over low heat and stir occasionally while frying the chicken.
Heat 3/4 to 1 cup canola oil (about 1–1½ inches) in a large frying pan over medium-high heat until hot for frying.
Place 1/2 cup cornstarch in a large resealable plastic bag and add the chicken pieces; shake to coat evenly.
Beat 2 eggs in a bowl and place 1/2 cup all-purpose flour in a separate bowl. Dip each cornstarch-coated chicken piece into the beaten eggs, then dredge in flour to coat.
Fry the coated chicken in the hot oil for 2–3 minutes per piece, turning with tongs, until golden, crispy, and cooked through. Transfer to a paper towel–lined plate; reserve a few tablespoons of oil in the pan and discard the rest.
Add the red and green bell peppers and pineapple chunks to the pan with the reserved oil and cook for 1–2 minutes until peppers are crisp-tender.
Finish the sauce: whisk the cornstarch slurry (3 tablespoons cornstarch mixed with 1/4 cup water), add it to the simmering sauce, bring to a low boil, and cook, stirring, until thickened about 2 minutes.
Add the thickened sauce to the pan with peppers and pineapple, then return the fried chicken to the pan and stir to coat evenly; cook until sauce is bubbling, about 4–5 minutes.