Place chicken in a large bowl and toss with 1/3 cup cornstarch, 1 tablespoon soy sauce, salt, and pepper, until evenly coated.
Heat 2 tablespoons vegetable oil in a large skillet or wok over medium-high heat until shimmery. Working in batches, add chicken in a single layer and sear until chicken is golden brown. Repeat cooking with remaining chicken, adding additional 1-2 tablespoons oil to pan as needed. Transfer chicken to a plate and set aside.
Heat remaining 1 tablespoon oil in same pan over medium-high heat. Add broccoli, sweet pepper, and green onions to pan and cook 2-4 minutes until tender. Add garlic and ginger and cook 1 minute until fragrant. Remove from heat for a moment.
In a small bowl, whisk remaining 1 teaspoon cornstarch, 1/3 cup soy sauce, pineapple juice, brown sugar, and rice wine vinegar until smooth.
Return pan with vegetables to stovetop and set over medium-high heat. Add chicken, sauce mixture, pineapple, and cashews to pan and toss over medium-high heat 2-3 minutes until sauce is thickened and coating vegetables and chicken.
Season stir fry with additional salt and pepper to taste if needed. Serve stir fry over steamed rice with additional green onions and enjoy!