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.
Heat oil in a wok skillet over medium-high heat. (Add more oil as necessary during cooking.) Cook and stir garlic in the hot oil for 15 seconds. Add broccoli, carrot, and onion. Cook and stir for 3 minutes. Add mushrooms; cook and stir for 1 to 2 minutes more or until vegetables are crisp-tender. Remove vegetables from wok with slotted spoon. Stir soy sauce mixture. Add to wok; cook and stir until slightly thickened and bubbly. Add shrimp or scallops; cook about 3 minutes or until shrimp or scallops are opaque. Stir in vegetables; heat through. Serve with rice or pasta and sprinkle with cashews.