Step 1: Cook the Noodles
1. Prepare Noodles: Cook the noodles according to package instructions. Drain and rinse under cold water to stop cooking. Set aside.

Step 2: Cook the Beef
1. Heat Oil: In a large skillet or wok, heat 1 tablespoon of vegetable oil over medium-high heat.
2. Stir-Fry Beef: Add the sliced beef in a single layer and cook for 2-3 minutes until browned. Avoid overcrowding the pan; cook in batches if necessary. Remove the beef from the pan and set aside.

Step 3: Stir-Fry Vegetables
1. Add More Oil: In the same skillet, add the remaining tablespoon of oil.
2. Add Garlic and Ginger: Sauté the minced garlic and ginger for about 30 seconds until fragrant.
3. Add Vegetables: Add the mixed vegetables and stir-fry for 3-4 minutes until they are tender-crisp.

Step 4: Combine and Sauce
1. Return Beef: Return the cooked beef to the skillet with the vegetables.
2. Add Noodles: Add the cooked noodles to the skillet.
3. Add Sauces: Pour in the hoisin sauce, soy sauce, rice vinegar (or lime juice), and sesame oil (if using). Toss everything together until well combined and heated through, about 2-3 minutes.

Step 5: Serve
1. Garnish: Serve hot, garnished with chopped green onions and sesame seeds.