Bring a large pot of salted water to a boil over medium-high heat. Meanwhile, make the sauce and prep the vegetables.
Stir 3 tablespoons soy sauce, 1 teaspoon packed brown sugar, and 1 teaspoon sesame oil together in a small bowl until the sugar is dissolved to make the sauce; set aside.
Thinly slice 1/2 medium yellow onion, cut 1 large broccoli crown into small 1-inch florets (about 5 cups), and mince 2 garlic cloves. Thinly slice 1 scallion for garnish if desired.
Add 8 ounces lo mein or udon noodles to the boiling water and cook according to package directions. Meanwhile, pat 1 pound cleaned shrimp dry with paper towels and season with 1/4 teaspoon kosher salt. Heat 1 tablespoon of the vegetable oil in a large nonstick skillet or wok over medium-high until shimmering. Add the shrimp in a single layer, and sear until browned on the bottom, about 2 minutes (they will not be cooked through). Transfer to a large plate and take the pan off the heat if the noodles aren’t ready yet.
When the noodles are ready, drain in a colander and run under cold water until cooled. Set aside to drain while you cook the vegetables.
Heat the remaining 2 tablespoons oil in the pan over medium-high heat until shimmering. Add the onion, season with salt, and stir-fry until starting to brown, 2 to 3 minutes. Add the broccoli and garlic, season with salt, and toss to combine. Add 2 tablespoons water, cover, and cook, stirring every minute or so, until the broccoli is bright green and crisp-tender, 2 to 3 minutes.
Uncover and add the noodles and reserved shrimp and any accumulated juices. Cook, tossing constantly, until the shrimp are just cooked through and the noodles are warmed through, about 2 minutes. Pour in the sauce and toss until it evenly coats the noodles and vegetables, about 1 minute. Garnish with the scallions if desired and serve.