Place block of tofu flat on cutting board; slice lengthwise into 4 equal pieces, then down the middle to make 8 squares. Place on baking sheet lined with three layers of paper towels; cover with three more layers. Place another baking sheet and a weight on top. Let tofu drain until towels are soaked, 5 minutes.
Spread the sesame seeds on a plate. Press both sides of each tofu square into sesame seeds. In a large nonstick skillet, heat 1 tablespoon sesame oil over medium heat. Cook tofu, until golden brown, 4 to 6 minutes per side. Add the soy sauce; continue cooking, turning tofu, until it has absorbed all the liquid, about 1 minute. Remove from pan. Wipe skillet with a paper towel.
Add broccoli, 1/4 teaspoon salt, 1/8 teaspoon pepper, and 1/2 cup water to skillet. Simmer, covered, until broccoli is tender, about 5 minutes. Divide among 4 serving plates; arrange 2 tofu squares on each. Drizzle with remaining tablespoon sesame oil. Serve.