Whisk all the ingredients for the peanut sauce together in a medium sauce pan. Bring to a simmer. Simmer 5 minutes, whisking occasionally until it thickens.
Heat a separate frying pan or wok and add 1 tablespoon vegetable oil. Add the broccoli florets and sprinkle with ½ teaspoon salt. Stir fry over high heat 3-4 minutes until the broccoli is bright green and crisp-tender. Remove broccoli to a platter.
Using the same pan, add 1 tablespoon oil, ½ teaspoon salt, and 1 teaspoon garlic to hot pan. Add chicken breasts and stir fry 3-4 minutes until the chicken is cooked through. Remove chicken to platter.