Stir together lime juice, water, and next 4 ingredients (through pepper) in a cup; set aside.

Heat a large, heavy skillet (such as cast iron) over high heat; lightly coat with cooking spray. Working in batches, if necessary, add steak to hot skillet in single layer; cook, without turning, until browned on 1 side and almost fully cooked (1-2 minutes). Transfer to a plate.

Heat oil in skillet over high heat. Add snap peas and cook until bright green and beginning to blister (2-3 minutes). Add jalapeño, peanuts, reserved beef (with juices), and reserved sauce to pan; cook until beef is done (about 1 minute). Toss with mint. Serve over brown rice, if desired.