Whisk the broth, soy sauce, and sesame oil together in a small bowl until combined.

In a separate bowl, stir the garlic, ginger, and the smaller portion of vegetable oil together until the aromatics are evenly coated.

Heat the remaining vegetable oil in a 12-inch nonstick skillet over high heat until it just begins to smoke.

Add the eggplant and cook, stirring frequently, until the edges are browned, about 3 minutes.

Push the eggplant to the edges of the skillet to clear the center, then add the garlic-ginger mixture; mash it into the pan with a spatula until fragrant, about 30 seconds.

Stir the aromatics into the eggplant, then pour in the broth mixture and immediately cover the pan.

Reduce the heat to medium and cook until the sauce has thickened to a glaze and the eggplant is fork-tender, about 3 minutes.

Transfer to a platter and garnish with scallions. Serve hot with rice.