Bring a large pot of water to a rolling boil and add your wide rice noodles, cooking them according to the package instructions. About 5 minutes before the noodles are done, toss the broccoli florets directly into the same pot so they cook together. Once tender, drain both the noodles and broccoli thoroughly in a colander and set them aside.
While your pasta water is working, heat the olive oil in your skillet over medium-high heat until it shimmers. Add the thinly sliced flank steak to the hot pan in a single layer to ensure it gets a nice sear. Cook the meat just until it is no longer pink, which keeps it tender and juicy.
Once the beef has lost its pink color, push it slightly to the side or mix in the minced garlic and sliced mushrooms. Sauté these together with the beef for about one more minute until the garlic is fragrant. This step infuses the meat with aromatic flavor and softens the mushrooms slightly.
In a small mixing bowl, combine the brown sugar, soy sauce, hoisin sauce, sesame oil, ginger, red pepper flakes, pepper, and cornstarch. Use a whisk to blend everything vigorously until the cornstarch is completely dissolved and the mixture is smooth. This preparation ensures the sauce thickens immediately when it hits the heat.
Add the drained noodles and broccoli into the skillet with the cooked beef and mushrooms. Pour your prepared sauce over everything and use tongs to toss the mixture until every strand is coated. Let it cook for 2-3 minutes until the sauce bubbles, thickens, and heats everything through.