Combine broccoli and water in a glass bowl; steam in microwave oven until slightly tender, 2 to 3 minutes.
Heat peanut oil in a large skillet or wok over medium-high heat. Cook garlic in hot oil until fragrant, about 1 minute. Reduce heat to low; add chicken broth, soy sauce, oyster sauce, and ginger root to the garlic. Bring the mixture to a boil and add the shrimp; cook and stir until the shrimp turn pink, 3 to 4 minutes. Toss steamed broccoli and water chestnuts with the shrimp mixture to coat with the sauce. Stir cornstarch into the mixture 1 tablespoon at a time until the sauce thickens, about 5 minutes.
