Combine the steak, 1 tablespoon cooking oil, 2 teaspoons soy sauce, and 2 tablespoons cornstarch in a large resealable plastic bag; seal and refrigerate to marinate for 1 hour.
When ready to cook, heat the remaining oil (about 1/4 cup) in a large skillet or wok over medium-high heat.
Dredge the marinated beef in the remaining cornstarch (use remaining 3 tablespoons), shaking off excess, and sauté in batches until crispy, about 2 minutes per side; transfer to a paper towel–lined plate.
Reduce heat to medium and add the garlic and ginger to the skillet; cook until fragrant, about 1 minute.
Stir in the remaining soy sauce (1/4 cup plus 2 teaspoons), water, brown sugar, and hoisin sauce; bring to a simmer and cook until slightly thickened, about 4–5 minutes.
Add the broccoli, cover the skillet, and cook until the broccoli is tender but still bright green, about 3 minutes.
Add the scallions and return the cooked beef to the skillet; toss everything in the sauce until evenly coated and heated through.
Serve the Mongolian beef and broccoli over cooked jasmine rice and garnish with sesame seeds.