Heat 1 teaspoon of vegetable oil in a large skillet over medium-high heat.
Add the broccoli, mushrooms, and bell peppers to the pan. Sauté for 5-6 minutes until they are tender but crisp. Remove from the skillet and cover to keep warm.
Wipe the pan clean, then heat the remaining tablespoon of vegetable oil over high heat.
Add the chicken pieces in a single layer. Season with salt and pepper, and cook for 3-4 minutes per side until golden brown and fully cooked.
Reduce the heat to medium, then stir in minced garlic. Cook for about 30 seconds until fragrant.
Return the sautéed vegetables to the pan.
In a small bowl, whisk together chicken broth, brown sugar, soy sauce, sesame oil, and cornstarch until smooth.
Pour the sauce mixture over the chicken and vegetables. Bring to a simmer and cook for 1-2 minutes until the sauce thickens.
Serve your garlic chicken stir fry hot, with steamed rice or cauliflower rice, if desired.