Preheat the oven to 400 degrees F. Grease a sheet pan with non-stick cooking spray.
Pat the tofu with a towel. Wrap it with a dry towel and place a cast iron pan or cutting board on top to remove more liquid. Let stand for 15 minutes. Cut the tofu into 1-inch cubes.
In a small saucepan over medium heat, combine the soy sauce, 1/4 cup water, maple syrup, mirin, ginger and garlic. Simmer the mixture for a couple minutes until it is fragrant. In a small bowl, stir together the cornstarch and 1 tablespoon water and then whisk into the sauce. Continue simmering until the sauce thickens, about 3-4 minutes.
Arrange the tofu in a single layer on half of a sheet pan and bake for 10-12 minutes.
Toss the broccolini with vegetable oil, salt and pepper.
Flip over the tofu. Add the broccolini to the other side of the pan. Continue roasting until the tofu is lightly browned and the broccolini is bright green and tender but still a bit crisp, about 10 minutes.
On the sheet pan, toss the tofu and broccolini with half the teriyaki sauce. Extra sauce can be served on the side or saved for another use.
Sprinkle with scallions and sesame seeds.
Serve with brown rice and the extra sauce on the side.