Cook the Udon: Begin by bringing a large pot of water to a boil. Add the udon noodles and cook according to package instructions (usually about 4-5 minutes). Drain and rinse with cold water to stop the cooking process, then set aside.
Prepare the Veggies: In a large skillet or wok, heat the sesame oil over medium-high heat. Add the minced garlic and grated ginger, stir-frying for about 30 seconds until fragrant.
Stir-Fry the Vegetables: Add the broccoli, red bell pepper, and carrot to the skillet. Stir-fry for about 4-5 minutes or until the vegetables are tender yet still crisp.
Add the Noodles: Reduce the heat to medium and add the cooked udon noodles to the skillet, tossing them gently with the vegetables.
Season the Dish: Pour the soy sauce and hoisin sauce over the noodle mixture, tossing everything to ensure even coating. Add the chopped green onions and sprinkle in the sesame seeds and red pepper flakes if using. Stir-fry for an additional 2-3 minutes to heat everything through.
Serve: Once well combined and heated, remove from heat and transfer to serving plates.
Notes
Garnish with fresh cilantro and sesame seeds. Serve with lime wedges for added zest.