Cook the udon noodles as per the instructions on the package. Fresh udon noodles don't have to be cooked long; you're looking at a cook time of about 2 minutes. When they're done cooking, strain them and rinse them with cool water, then set them aside.
Heat 2 TBSP of sesame oil in a large wok or skillet. Once the oil has heated up, add the garlic, mushrooms, and broccoli. Give everything a quick toss and stir so that the garlic doesn't burn. Sauté these ingredients until tender, stirring occasionally. This should only take about 3 minutes. If you're wok seems to dry out, carefully add a splash or two of warm water to it. This helps loosen things up and prevents sticking and burning.
Once the veggies are tender, add the carrots and scallions. Stir the veggies to combine them and sauté for 1 minute.
Finally, add the udon noodles to the pan. If the udon noodles seem to be sticking together before adding them to the pan, rinse them under cool water again to loosen them up beforehand.
Once you'd added the udon noodles to the pan, pour the sauce mixture over the contents in the pan and give everything a toss until all of the ingredients are evenly coated.
Plate the noodles and veggies, and garnish with your choice of garnishes. (Tip: if you're adding bean sprouts, add them to the wok when you add the udon noodles. This way they warm up a bit without losing their crisp.)