To prepare the meat: In a small measuring cup, combine lime juice, 2 tablespoons fish sauce, 3 tablespoons brown sugar and chile. Whisk until sugar is dissolved. Transfer half of the mixture into a medium bowl and the remaining mixture into a large zip-top bag.
Add remaining 2 tablespoons fish sauce and garlic to the zip-top bag. Shake to combine. Add steak, seal bag and turn several times to coat steak in marinade. Keep refrigerated for at least 30 minutes or up to 8 hours.
To make the salsa: In the bowl with the lime juice mixture, add the cucumber, mango, cilantro and mint. Stir to combine. Set aside.
To make the Peanut Sauce: In a small bowl, whisk together the peanut butter, 2 tablespoons lime juice, water, Sriracha sauce, 1 tablespoon brown sugar and 2 teaspoons fish sauce. Set aside. (Add more water, a tablespoon at a time, if the sauce is too thick to drizzle.)
To complete the dish: Preheat your broiler to high and place rack 6 inches from broiler. Place a piece of foil over a large baking sheet and lay the skirt steak on top. Broil for about 4 minutes on each side (for medium-rare); broil a bit longer on each side if you prefer it more well done. Transfer to a cutting board and let rest for 10 minutes. Cut into thin slices against the grain.
Place a slice or two of meat in each warm tortilla, drizzle with peanut sauce and top with sliced cabbage, cucumber and mango salsa, cilantro and mint leaves and serve.