Press the tofu between paper towels or a clean kitchen towel with a heavy pan on top for 15 minutes to remove excess moisture. Cut into 1-inch cubes and toss with cornstarch until lightly coated.
Heat 2 tablespoons sesame oil in a large skillet over medium-high heat. Add the tofu cubes in a single layer and cook, turning occasionally, until golden and crisp on all sides, about 8 minutes. Remove tofu to a plate.
Add the remaining tablespoon of sesame oil to the same skillet. Stir in the garlic and ginger and cook for 30 seconds until fragrant.
Add the broccoli florets and cook, stirring often, for 4 minutes until bright green and starting to soften.
Whisk together the soy sauce, maple syrup, and rice vinegar. Pour the sauce into the skillet with the broccoli and stir to coat. Return the tofu to the pan and toss everything together for 2 minutes until heated through.
Remove from heat and sprinkle with sesame seeds. Serve the stir-fry over cooked rice and top with sliced green onions.