Combine the clam juice, soy sauce, sesame oil, vinegar, cornstarch, sugar, chile sauce, and 1/2 tsp. pepper in a glass measuring cup or medium bowl.
Heat 1-1/2 Tbs. of the vegetable oil in a 12-inch nonstick skillet over medium-high heat. Working in batches, cook the eggplant in a single layer until almost tender and starting to brown, about 3 minutes per side. Transfer each batch to a paper-towel-lined plate when done and add more oil between batches as needed.
Heat 1-1/2 Tbs. of the oil in the same skillet over medium-high heat until very hot. Lightly sprinkle the scallops with salt and pepper and cook until brown and almost cooked through, 2 to 3 minutes per side. Transfer to a plate.
Add the remaining 1/2 Tbs. oil and the scallions, ginger, and garlic to the skillet; stir until fragrant, about 30 seconds. Add the Broccolini, sprinkle with salt, and stir until heated through, about 1 minute. Cover and cook until crisp-tender, stirring occasionally, 6 to 8 minutes. Stir the broth mixture to combine and add to the skillet. Bring to a simmer, stirring constantly. Add the eggplant and the scallops and simmer just until heated through, 2 to 3 minutes. Serve.