Using a food processor or immersion blender, combine Fresh Basil Leaf (1/2 cup), Garlic (3 clove), Lemon (2 tablespoon), Pine Nuts (1/8 cup), Salt (1 teaspoon), Ground Black Pepper (1 teaspoon), and Olive Oil (1/4 cup).
Add the Egg (2). Process until sauce thickens slightly, about 1-3 minutes.
Stream in Vegetable Oil (1/2 cup) until sauce thickens. Set aside and refrigerate until ready to use.
Cut Eggplant (1) into 1-inch thick rounds.
In a large bowl, combine Olive Oil (3/4 tablespoon), Balsamic Vinegar (3/4 tablespoon), Ground Cumin (1/2 teaspoon), Garlic Powder (1/2 teaspoon), Ground Black Pepper (1 teaspoon), and Salt (1 teaspoon) together.
Toss eggplant in the mixture to coat.
Slice the Halloumi Cheese (4 ounce) into 1/2-inch thick slices.
Heat a grill pan over medium-high heat. Once hot, cook the eggplant for 6 minutes on each side.
In the same pan grill, cook the cheese for 2 minutes on each side or until grill marks appear.
Assemble the burger. Spread aioli on 2 halves of a burger Buns (4). On the bun, stack eggplant, halloumi, and top with Tomato (1) and Lettuce (1/4 head).