Prep: Give your choice of vegetable a good scrub and pat dry. If leaving the vegetable whole, move on to tenderizing. If cutting, decide on the approximate shape and size desired. Cut pieces similar in size to ensure even grilling.
Tenderize: Once you've prepped the vegetables, choose a method of tenderizing if needed. For vegetables higher in water content, consider salting and resting for up to 30 minutes. If brining, plan to start this a few hours or more before grilling.
For larger vegetables that need an initial cook, choose to parboil, steam, or blanch as needed. I prefer to steam vegetables that might have delicate pieces, such as a wedges of cauliflower. Heartier vegetables, such as potatoes, are acceptable to boil (albeit at a temperature slightly below a boil so the outside of the potato doesn't fall apart). After precooking, place in an ice bath to prevent further cooking.
If planning to make skewers, thread the vegetables onto the skewers once cool.
Preheat your grill: Once your vegetables are ready, preheat your grill. I typically grill at around 400˚F to 450˚F for direct and indirect heat. If looking to cook directly on the coals, let the grill burn until all the coals are embers and the internal temperature is between 200˚F and 300˚F.
Cook: Once your grill is ready, brush/drizzle your vegetables with enough oil to coat them. Place them on the grill or in a grill basket. Cook, turning occasionally, until you meet your desired level of char.
Once charred, either remove the vegetables from the grill if done or move to indirect heat to finish cooking. Once all the vegetables finish, remove them from the heat and either cover them to keep cooking with residual heat or be ready to add flavor.
Finish them: Once the vegetables are grilled and ready, toss with your desired sauce, more fats, herbs, spices, something acidic, or more salt as needed.