Rotisserie chicken and alllll the spices make ramen even better.
For the sauce, in a small saucepan, bring all of the ingredients to a low bubble over medium heat. Cook, stirring often, until the sauce thickens a bit, about 5 minutes.
Bring a large pot of water to a boil for the noodles.
In a large skillet, heat 1 tbsp. oil, one turn of the pan, over medium. Add the eggs. Cook, stirring with a spatula, until lightly brown and scrambled, about 1 minute. Transfer the eggs to a plate.
Add another 1 tbsp. oil, one turn of the pan, to the skillet. Add the chicken, sprinkle with the five-spice powder, and cook until heated through, about 3 minutes. Add the mirin and cook until the liquid is absorbed, about 1 minute. Transfer the chicken to another plate. Add the remaining 1 tbsp. oil, one turn of the pan, to the skillet. Add the mushrooms, pepper, and carrot. Cook, stirring often, until the vegetables soften, about 2 minutes.
Add the noodles to the boiling water and cook just until tender, 2 to 3 minutes; drain.
Add the sprouts and scallions to the skillet with the vegetables. Add the noodles, egg, chicken, and sauce; toss to coat. Serve the lo mein from the pan topped with the sesame seeds, if using.