Combine the chicken, cornstarch, measured salt, and pepper in a large bowl and toss until the chicken is lightly coated; set aside.
Heat the oil in a small frying pan over medium-high heat until shimmering. Add the scallions, garlic, and ginger, season with a pinch of salt, and cook, stirring until bright green and fragrant, about 1 minute. Scrape into a 5-quart or larger slow cooker.
Add the chicken broth, peanut butter, soy sauce, red curry paste, and sugar. Whisk together until smooth and well-combined. Add the chicken, scraping out any cornstarch stuck to the sides of the bowl, and stir to combine. Cover and cook on the LOW setting without stirring until the chicken is cooked through and tender, 4 to 6 hours.
Add the broccoli, gently stir to combine, and cover and cook until the broccoli is crisp-tender, 30 to 45 minutes more. Serve over rice, topped with chopped peanuts and scallions. Serve the chili-garlic paste on the side.