Bring a pot of water to a boil for the noodles. One it’s boiling, add the noodles - they only take about 3 to 5 minutes. Discard any flavor packets. Once the noodles are done, drain them, reserving a few tablespoons of the pasta water.
While the water is coming to a boil for the noodles, heat the olive oil in a large skillet over medium heat. Add in the mushrooms and stir. Cook for 5 minutes, until the mushrooms soften a but. Stir in the garlic, carrots and green onions (I reserve a few for topping). Cook for 1 to 2 minutes. Stir in the soy sauce, toasted sesame oil, honey and vinegar.
Add the noodles right to the skillet and toss well. If you’d like, stir in 1 to 2 tablespoons of the reserved pasta water. Toss occasionally while you make the eggs for topping.
To make the eggs, heat 1 to 2 tablespoons of toasted sesame oil in a separate skillet over medium heat. Once hot, crack the eggs right into the oil. Season with a pinch of salt and pepper. Let the egg bubble and cook in the oil. If desired, you can spoon a but of oil over the yolk and whites to cook it quicker. I find that this takes 2 to 3 minutes total to cook and get a solid white and runny yolk - depends on how done you like your egg!
Serve the veggie ramen immediately topped with an egg. Sprinkle on any extra green onions and sesame seeds