Preheat oven to 400 degrees F. Line 2 baking sheets with parchment paper; set aside.
Drain water from the tofu. Cut tofu width-wise to create 2 thinner slabs of tofu. Gently layer between layered paper towels or a clean kitchen towel, gently pressing out excess water. Cut into ¾-inch cubes and arrange in a single layer on one of the prepared baking sheets. Mist with olive oil cooking spray and sprinkle with ¼ teaspoon salt and pepper. Bake for 15 minutes.
Meawhile, place cabbage and broccoli on the second baking sheet. Drizzle with olive oil and the remaining ½ teaspoon salt and black pepper, to taste. Gently toss and arrange in a single layer.
Once the tofu has baked for 15 minutes, add the sheet pan of vegetables, baking for 15 minutes while tofu bakes an additional 15 minutes.
Meanwhile, prepare the peanut sauce by mixing together all ingredients in a bowl or small mason jar.
Combine the tofu and vegetables in a large bowl; drizze with peanut sauce and gently toss to coat. Sprinkle with sliced green onions and sesame seeds. Serve over steamed brown rice or noodles, if desired.*