Heat the oil in a skillet and cook the onion, garlic and red bell pepper for 2-3 minutes. Allow to cool slightly.
Meanwhile, in a mixing bowl put the ground turkey, cream cheese, breadcrumbs, fresh basil and cheddar cheese. Season to taste with salt and pepper. Add the pepper and onions and mix well.
Divide the mixture into four and shape into burgers. If you have time, cover and chill in the fridge for 20 minutes to let the flavours blend (I skipped this step to save time).
Place a sheet of foil over the barbecue or grill, or use a baking sheet if preferred- because these burgers are made from a low fat meat, they can break up if not handled carefully, so the foil helps them to stay together. Handle and flip them with care.
Place the burgers on the barbecue or under a grill, on the foil or baking sheet.
Cook for 15 minutes, flipping regularly.
If using a barbecue, remove the foil at this point- after 15 minutes- so the burgers pick up the chargrilled flavour in the next 10 minutes.
Cook for a further 10 minutes (whether using a barbeque or a grill) or until completely cooked through.
Prepare the rolls and serve the burgers with lettuce, tomato or your favourite toppings.