Prepare the vegetables: Wash and dry the broccoli and green beans. Trim the beans, cut them into bite-sized pieces, and separate the broccoli into evenly sized florets.
Heat the oil: Warm the olive oil in a large skillet over medium heat.
Cook the onion: Add the sliced onion and sauté for about 3 minutes, stirring occasionally, until softened.
Start the green beans: Add the green beans and cook for 2–3 minutes, stirring to coat them in the oil.
Add the broccoli: Stir in the broccoli florets and 2 tablespoons of water. Cover the skillet and cook for 5–7 minutes, checking occasionally for tenderness.
Add the aromatics: Remove the lid and allow excess moisture to evaporate. Stir in the minced garlic and grated ginger.
Season the vegetables: Add the soy sauce or tamari and cook for another 1–2 minutes, stirring frequently to prevent the garlic from burning.
Finish with lime: Turn off the heat, add lime juice, and taste before adjusting the salt and pepper.
Garnish and serve: Sprinkle with sesame seeds if desired and serve warm with your favorite main dish.