For the grilled baby eggplant: Preheat a grill or grill pan to medium high.
Cut off the stems from the baby eggplant and slice them in half lengthwise. Make a 1/2-inch-deep crosshatch pattern on the flesh side of each using a knife. Put them in a medium bowl.
In a small bowl, mix together the oil, rosemary and garlic. Pour the mixture over the eggplant, tossing to coat completely, and sprinkle generously with salt and pepper. Grill, flipping halfway through, until tender and dark, about 10 minutes.
For the glaze: Meanwhile, in a small saucepan, combine the oil, stock, balsamic vinegar, honey, mustard, lemon juice, salt and pepper. Cook over low heat until the glaze is reduced by half, about 10 minutes. Drizzle over the eggplant to serve.