Heat 1 teaspoon of the oil in a large pan over medium high heat. Add the broccoli, peppers and carrots, and cook for 4-6 minutes until softened and lightly browned. Season to taste with salt and pepper.
Remove the vegetables from the pan; place on a plate and cover with foil to keep warm.
Add the remaining 2 teaspoons of oil to the pan. Add the pork, then season with salt and pepper to taste. Cook, stirring occasionally, until the meat is browned and cooked through, 4-6 minutes.
Add the garlic and ginger to the pan and cook for 30 seconds.
While the pork is cooking, make the sauce. Place the soy sauce, water, brown sugar, honey and sesame oil in a small pot over medium high heat. Stir until sugar is dissolved, about 3 minutes. Turn up heat to high and bring to a boil.
Mix the cornstarch with 2 tablespoons of cold water until dissolved. Add the cornstarch mixture to the sauce and boil for 1-2 minutes or until sauce has thickened.
Add the vegetables back to the pan with the pork. Pour in the sauce and toss to coat. Sprinkle with sesame seeds, then serve.