In a small bowl, whisk together 1/2 cup chicken broth, 1/4 cup soy sauce, 3 cloves minced garlic, 3 tablespoons brown sugar, 1 teaspoon sesame oil, and 1 tablespoon cornstarch and set aside.: The moment you whisk these together you should smell the warm garlic and the toasty hint from the sesame oil . The mixture should look uniform, with no visible lumps from the cornstarch . This preblended sauce ensures even thickening when it hits the hot pan. If you skip this step, the cornstarch can clump and leave grainy pockets, so always dissolve it fully. Keep the bowl close to the stove so you can add the sauce quickly while the pan is hot.
In a wok or large skillet, add 1 tablespoon vegetable oil over medium-high heat. Add 2 cups broccoli florets, 1 cup sugar snap peas, 1 sliced red bell pepper, and 1/2 cup shredded carrots. Cook for about 2-3 minutes, stirring occasionally.: When the oil shimmers, it is hot enough to sizzle the vegetables on contact. You should hear a lively sizzle and see bright colors intensify as the edges pick up slight browning. Stir intermittently to let each piece touch the pan then move it around, which helps develop tiny caramelized spots that add flavor. Avoid overcrowding the pan because that releases steam and results in limp vegetables. If the vegetables steam instead of sear, raise the heat slightly and work in batches.
Add 1 pound peeled and deveined shrimp and cook for 2-3 minutes until the shrimp begin to turn pink.: As the shrimp hits the hot pan you should hear a pronounced sizzle and see them start to opaquely change color at the edges. When they curl slightly and become mostly pink with a pearly center, they are approaching doneness. The shrimp should have a firm spring to the touch, not rubbery. Overcooking will tighten their texture, so remove heat promptly once they are just opaque.
Pour in the sauce and stir until incorporated. Cook for 3-5 minutes, stirring occasionally until broccoli is tender and the sauce has thickened. Garnish with sliced green onions and serve over rice if desired.: When the sauce hits the hot skillet it should instantly begin to bubble and thicken as the cornstarch activates. Stir to coat every piece so the glossy sauce clings to the shrimp and vegetables. The aroma will deepen, with sweet, savory notes becoming more pronounced. If the sauce becomes too thick, a splash of reserved chicken broth will loosen it. Avoid adding cold water directly, because that can cool the pan and slow thickening.
Cook for 3 to 5 minutes, stirring occasionally until broccoli is tender and the sauce has thickened: During these final minutes watch the sauce develop a sheen and the vegetables reach tender crisp. You want the broccoli florets to yield with a fork but still hold shape, and the shrimp to feel springy. Stir occasionally so nothing sticks and the sauce reduces to a coatable consistency. A common mistake is to rush this step, which can leave the sauce thin and the flavors underdeveloped, so allow the pan to work for a few minutes while you observe color, aroma, and texture coming together.
Garnish with sliced green onions and serve over rice if desired: The final touch of sliced green onions adds freshness and a mild bite. As you plate, notice how the glossy sauce clings and how the colors pop, which signals balanced cooking. Serve promptly for the best texture, because the vegetables will continue to soften as they cool. Reheating later is fine, but expect a modest loss of crispness in the vegetables.