Cut your broccoli into florets & small cubes of broccoli stem, keeping the two parts separate and drying them to get off any residual water. Mince both the ginger and the garlic, zest the lime, and dissolve the salt into 2 Tablespoons of water.
Turn on medium-low heat under a cast iron* or nonstick pan, and add your sesame oil, giving it 10 seconds to heat up a bit (any longer and it will pop when you add your ingredients).
Add a third of your garlic, half of your ginger, and all your broccoli stems to the pan, heating for 1 minute.
Add all the broccoli florets and stir until everything has a light coating of oil, then add the rest of the ginger & garlic and stir everything again until well-distributed.
Arrange the broccoli mixture in a single layer in the pan, cooking it for 9 more minutes & stirring every 30 seconds or so. You want to encourage as many pieces of broccoli to brown and crisp up, but NOT burn. If anything begins to burn, reduce the heat and stir until the burning bits are on top of yet-to-be-cooked bits— the ginger and garlic should also brown but not burn.
Once fully cooked (a fork goes easily into the stem of the largest piece of broccoli), pour in your tahini and honey and stir until evenly-coated, then increase heat to medium high and cook all together for 30 more seconds.
Once the pan has heated up to temperature, which may take another 30 seconds, pour on the salt water and stir immediately and constantly, until the coating on the outside is more of a sauce (about 1-2 minutes).
Then turn off the heat and add your lime zest and sumac, stirring until evenly-coated, and then sprinkle toasted sesame seeds on top before serving.