Cook the tofu: Cut the tofu into bite sized pieces (1-inch squares, 3/8-inch thick and pat it dry with a towel. Add 2 tablespoons sesame oil to a large non-stick pan and add the tofu cubes and a few pinches kosher salt. Turn on the heat to medium-high. Cook 5 to 6 minutes until lightly browned on the bottom.
Flip the tofu: Briefly remove the pan from the heat to reduce spitting. Flip the tofu with chopsticks (the easiest method!) or tongs. Return the heat to medium-high and cook 5 to 6 minutes until browned. Remove the tofu to a bowl and set it aside.
Stir fry the veggies: Add the remaining 1 tablespoon oil to the skillet. Add the broccoli, pepper, and shiitake mushrooms and two pinches salt and cook 6 to 7 minutes until just starting to brown on edges, stirring occasionally. In the last minute, add 1 tablespoon water and allow the vegetables to steam.
Turn down the heat to low. Add the green onion and ginger and cook 1 minute. Turn off the heat. Add the tofu and sauce and stir until the sauce thickens. Garnish with sesame seeds if desired. Serve immediately. Store leftovers up to 3 days refrigerated.