Preheat the oven to 375 degrees Fahrenheit. Line and spray a baking sheet with nonstick cooking spray. Place eggplant and zucchini sticks on baking sheet and sprinkle with salt and pepper.
In a small mixing bowl, whisk together sesame oil, olive oil, and garlic. Drizzle over the vegetables. Bake for 10–12 minutes until soft but still firm.
Meanwhile, prepare the sauce. In a small saucepan, combine vinegar, salt, honey, soy sauce, garlic, water and cornstarch. Mix well and let simmer over low heat until thickened. Pour over zucchini and eggplant sticks and serve immediately.