In a small bowl, whisk together the lime juice, soy sauce, rice vinegar, sesame oil, and brown sugar until the sugar dissolves completely. Set this sauce mixture aside.

Heat a wok or large skillet over medium-high heat. Add a splash of sesame oil and sauté the minced garlic for about 30 seconds until fragrant, being careful not to burn it.

Add the shrimp to the pan and cook for 2-3 minutes, turning occasionally, until they turn pink and opaque. Remove the shrimp from the pan and set them aside.

In the same pan, add a bit more sesame oil if needed. Toss in the chopped red onion and broccoli florets. Stir-fry for 4-5 minutes until the vegetables are tender-crisp.

Return the cooked shrimp to the pan with the vegetables. Pour the prepared sauce over the mixture and toss everything together to coat evenly. Cook for an additional 2 minutes, allowing the sauce to thicken slightly. Garnish with chopped green onions before serving.