Char onions for flavor, saute them for texture
I thinly slice onions in rings and grill them on high heat for 5-8 minutes until they develop a char and become slightly tender. Then I transfer them to a large skillet (on either the stovetop or grill) and caramelize them in olive oil with a drizzle of honey, pinch of salt, and cayenne pepper. Cook the rings over low heat until they're tender, about 25-30 minutes. I serve onions like this with steak, shrimp, and burgers, or even on a cauliflower steak.
Eggplant needs help in the oven before it's transferred to the grill
First I halve the eggplant lengthwise and arrange the halves flesh-side up on a baking sheet. Then I drizzle them with olive oil, then toss on some dried oregano, paprika, and salt. Roast them in the oven until tender and browned, about 15-30 minutes, depending on the size of the eggplant. Place the eggplant halves flesh-side down on the hot end of the grill and cook for an additional 5-10 minutes or until they develop a nice char. I like to finish them with a sprinkle of balsamic vinegar. This kind of eggplant preparation goes well with pasta and lamb and on pizza.