Toast the sesame seeds by first heating a small, stick-free sauté pan on medium high heat. Add the sesame seeds and jiggle the pan so they spread out in a single layer.
Let them cook until lightly browned, stirring occasionally, about 2-3 minutes. Do not walk away from them while cooking, as once they start to brown they can easily burn. Once lightly toasted remove from heat and put into a small bowl.
Mix the stock, soy sauce, and dark sesame oil together in a small bowl, set aside.
Heat 1 Tbsp of peanut oil (or canola or other high smoke point oil) in a large, cover-able sauté pan on medium high heat. Add the broccoli florets, stir to coat the florets with the oil, sauté for about a minute.
Clear a space in the middle of the broccoli and add the ginger and garlic. Add a little more oil to the ginger and garlic (about a teaspoon) and sauté for half a minute, stirring just the garlic and ginger, until fragrant. Then stir the garlic and ginger in with the broccoli.
Add the stock soy sesame oil mixture to the pan. Bring to a simmer, reduce the heat and cover. Let cook for 2-3 minutes, until broccoli is still firm, but can be pierced with a fork. Remove from heat.
Remove broccoli with a slotted spoon to a bowl. Return pan to heat, increase heat to high and boil down the liquid until just a couple tablespoons remain.
Turn off heat, return broccoli to the pan, add the toasted sesame seeds, toss with the liquid. Put into a serving bowl.