Bring 2 1/4 cups water and the ginger slices to a boil in a medium saucepan over high heat. Stir in the rice and return to a boil. Cover, reduce the heat to low and cook until the water is absorbed and the rice is tender, about 18 minutes. Fluff with a fork, cover and set aside.
Meanwhile, mix the chicken broth, ketchup, chili-garlic sauce, oyster sauce, cornstarch and 1/2 cup water in a measuring cup; set aside. Heat 2 tablespoons vegetable oil in a large nonstick skillet over medium-high heat. Add the shrimp in two batches and cook until no longer translucent, 1 to 2 minutes per side; remove to a large bowl.
Add the broccoli and 1/2 cup water to the skillet and cook, stirring occasionally, until crisp-tender and the water has evaporated, about 3 minutes. Add to the bowl with the shrimp.
Add the remaining 2 tablespoons vegetable oil to the skillet. Add the minced ginger and stir-fry until golden, 30 seconds to 1 minute. Pour in the prepared sauce and simmer until thickened, about 2 minutes. Add the shrimp and broccoli and toss to coat, adding a splash of water to loosen, if needed. Serve with the rice.