1. Sprinkle chicken with salt; toss with 1/4 cup cornstarch.
2. Stir-fry chicken in 3 Tbsp. hot oil in a large skillet or wok over medium-high heat 5 to 6 minutes or until golden brown and done. Transfer to a plate, using a slotted spoon; keep warm. Add Broccolini and 1/4 cup broth; cover and cook 1 to 2 minutes or until crisp-tender. Transfer to plate with chicken, using slotted spoon.
3. Add remaining 1 Tbsp. oil to skillet. Sauté bell pepper and next 2 ingredients in hot oil 2 minutes or until crisp-tender.
4. Whisk together 2 tsp. cornstarch and remaining 3/4 cup broth until cornstarch dissolves. Add broth mixture, chicken, and Broccolini (with any accumulated juices) to bell pepper mixture in skillet. Cook, stirring often, 1 minute or until liquid thickens. Stir in lime juice and next 2 ingredients. Serve over hot cooked rice.