Pat the flank steak dry with a paper towel, season with black pepper, and slice it against the grain into very thin, bite-sized strips.
In a small glass bowl, whisk together the tamari, rice vinegar, and freshly grated ginger until well combined to create the stir-fry base.
Heat the avocado oil in a large cast-iron skillet or wok over medium-high heat until it begins to shimmer.
Add the beef strips in a single layer and sear without moving for 2 minutes to develop a savory golden crust, then flip and cook for 1 more minute before transferring to a plate.
Lower the heat to medium and add the toasted sesame oil followed by the broccoli florets, sliced bell peppers, and sugar snap peas.
Stir-fry the vegetables for 3 to 4 minutes, adding the minced garlic in the final minute of cooking to ensure it becomes fragrant without burning.
Return the seared beef and any accumulated juices to the pan, then pour the tamari and ginger mixture over the top.
Toss everything together for 60 seconds until the sauce thickens slightly and creates a glossy glaze over the beef and vegetables.
Serve the stir-fry immediately over the warm brown rice and garnish with red pepper flakes and toasted sesame seeds.