Pre heat your grill to high heat, letting it heat up at least 10 minutes.
Heat the olive oil (for the veggies) in a large frying pan on medium high heat. Add in the pepper, onion, celery and garlic and cook, stirring frequently, until soft and golden brown, about 5-7 minutes. Keep on low heat to keep warm.
Combine the ground beef, Cajun seasoning and salt in a large bowl and mix evenly. Form into 3 3/4 inch patties. Brush each side lightly with olive oil.
Once the grill is on high heat, put the burgers over direct heat and cook for 4 minutes. Flip and cook an additional 3-4 minutes, or until they reach the desired level of done-ness. Transfer to a plate and cover to keep warm and let rest.
Make 8 slices out of the middle of the sweet potato (it's the largest part) about 1/4 inch thick. Place them into a large, microwave-safe bowl and add a little water. Cover and microwave until they JUST begin to soften, only about 2 minutes. Drain the water and pat the potatoes dry with a paper towel. *
Brush each side of the potatoes with the oil and grill until nice char marks form, about 4 minutes. Flip and cook and additional 3-4 minutes until fork tender, but not mushy.
While the potatoes cook, mix all the ketchup ingredients in a medium bowl.
Divide the ketchup between all 8 of the sweet potato slices. Divide the veggies between 4 of the slices, place one burger on top of each veggie-covered slice. Cover with the remaining sweet potatoes.
*I am sure you could boil them too, but it would only be for 2-4 minutes or so.