Heat 1 tablespoon olive oil in a large saute pan over medium high heat. Add chicken breasts, and season with a generous pinch of salt and pepper. Cook for about 5 minutes, stirring occasionally, until the chicken is browned and mostly cooked through.
While the chicken is cooking, make your sauce (see below).
Once the chicken is browned, add the remaining 1 tablespoon of olive oil and broccoli, and stir to combine. Continue cooking for an additional 3 minutes, until the broccoli is broccoli is bright green. Stir in the sauce and cook for an additional 1 minutes until the sauce has thickened. Remove from heat and stir in the sesame oil until combined.
Serve warm, garnished with green onions and toasted sesame seeds.
TO MAKE SAUCE:
Whisk all ingredients together in a small bowl until combined