Bring a large pot of water to a boil. Cook noodles according to package directions until just al dente, typically 6 to 8 minutes. Drain well and toss with a tiny drop of sesame oil to prevent sticking. Set aside.

In a small bowl, whisk together minced garlic, soy sauce, sriracha, honey, rice vinegar, sesame oil, grated ginger, and red pepper flakes until well combined. Set aside.

Heat vegetable oil in a large skillet or wok over medium-high heat until shimmering. Add thinly sliced chicken in a single layer. Cook undisturbed for 2 to 3 minutes until golden brown on the bottom. Flip and cook another 2 to 3 minutes until cooked through and golden. Transfer to a plate if working in batches.

Add broccoli florets to the same skillet. Stir-fry for 3 to 4 minutes, tossing frequently, until bright green and crisp-tender. Add a tablespoon or two of water if needed to steam slightly.

Return all chicken to the skillet with broccoli. Pour the prepared sauce over everything. Toss to coat and simmer for 1 to 2 minutes, stirring constantly, until sauce thickens and becomes glossy.

Add cooked noodles to the skillet. Use tongs to toss everything together until noodles are fully coated and evenly distributed with chicken and broccoli. Continue tossing for 1 to 2 minutes until noodles are heated through.

Divide among four serving bowls. Top generously with sliced green onions and toasted sesame seeds. Serve immediately while hot.