If you have time, pop the steak in the freezer for 5 minutes; that will make the meat easier to cut.
Peel, trim and grate the carrots (on a box grater or in a food processor fitted with a shredding disk). Mince the garlic. Remove any loose outer leaves from the cabbage, then cut it in half and core it. Cut the cabbage halves into very thin slices. If using peanuts, coarsely chop them.
Stir the peanut butter to re-incorporate any oil on the top, if needed. Whisk together the peanut butter, juice, soy sauce, vinegar and sugar in a medium bowl until smooth.
Heat 2 teaspoons of the oil in a wok or large skillet over medium-high heat.
Meanwhile, trim any visible fat from the steak. Cut the meat crosswise into very thin slices.
Add the garlic to the hot oil and cook for 30 seconds, stirring so it doesn't burn. Add the steak and stir-fry for 2 to 4 minutes, until barely browned but not completely cooked through. Use tongs to transfer to a bowl.
Add the remaining 2 teaspoons of oil to the wok or skillet. When it is hot, add the cabbage and 2 tablespoons of the water. Stir-fry for 3 to 5 minutes so the cabbage starts to wilt. Add the carrots and the remaining tablespoons of water as needed to keep the vegetables from sticking.
Return the steak to the wok or skillet, along with any accumulated juices; stir to incorporate. Pour in the peanut sauce and stir-fry to heat through and coat evenly.
Divide among individual wide, shallow bowls; if you're serving this stir-fry on top of noodles, sprinkle each portion with 1/2 teaspoon of the toasted sesame oil before adding the stir-fry on top. Garnish with a sprinkling of the chopped peanuts, if using. Serve hot.