Roasted Eggplant Veggie Burgers

Name: Roasted Eggplant Veggie Burgers
Ingredients: 1 12-ounce eggplant, halved lengthwise; 3 tbsp. olive oil, divided; Kosher salt and pepper; 1 medium onion, chopped; 2 cloves garlic, chopped; 1 tbsp. paprika; 1 tsp. dried thyme; 1 tsp. dried oregano; 1/2 c. quick-cooking oats; 1/2 c. raw walnuts; 1 15-ounce can lentils, rinsed; 1 large egg yolk; 1 tsp. Worcestershire sauce; 4 slices American cheese (optional); 4 brioche buns; Iceberg or romaine lettuce, mayonnaise, tomato, red onion, and pickled jalapeño, for topping