Marinate the tofu: Whisk together the soy sauce, sesame oil and garlic-ginger paste in a large bowl. Add cubed tofu, toss to coat, and set aside to marinate while you prep everything else.
Make the stir fry sauce: Whisk together the soy sauce, dry white wine, oyster sauce, sesame oil, garlic-ginger paste and cornstarch in a medium bowl until cornstarch is fully dissolved. Set aside.
Cook the noodles: Bring a large pot of water to a boil. Cook the lo mein noodles according to package directions until just al dente. Drain and rinse well under cold running water until completely cool. Toss with a small drizzle of sesame oil to prevent sticking and set aside.
Steam the broccoli: Heat a large skillet over medium heat. Add the broccoli, cut into small bite-sized florets and ½ cup water, then cover with a lid. Cook 2 to 3 minutes until bright green and tender-crisp. Transfer broccoli to a bowl with a slotted spoon and discard remaining water.
Cook the tofu: Return skillet to medium heat. Add 1 tablespoon of neutral oil and heat until shimmery. Add marinated tofu and 2 tablespoons of marinade. Cook, stirring frequently, 3 to 4 minutes.
Add garlic and green onions: Add remaining 1 tablespoon of neutral oil, divided, minced garlic, and both parts of the green onions. Cook 1 to 2 minutes, stirring often, until fragrant.
Finish the stir fry: Reduce heat to medium. Add stir fry sauce, noodles, broccoli and shelled edamame. Toss to combine and cook 30 to 60 seconds until sauce has thickened and everything is coated. Remove from heat.
Chill: Transfer to a wide bowl or spread onto a sheet pan to cool quickly. Once at room temperature, cover and refrigerate at least 1 hour or until fully chilled.
Serve: Toss noodles before serving — add a small splash of soy sauce or sesame oil if they've dried out. Divide into bowls and garnish with 1 tablespoons sesame seeds, for garnish and additional sliced green onions.