Cook the portobellos four at a time: Heat 2 tablespoons of the oil in a nonstick pan over medium heat until hot. Season the portobellos with salt and pepper and cook until lightly browned, about 2 minutes per side. Remove the portobellos from the pan and set them on a paper-towel-lined plate.
Season each of the tomato slices with a pinch of sugar and sear them in the pan until lightly browned, about 1 minute per side. Add the turkey slices to the pan and sear until they start to brown, about 10 seconds per side. Set the tomato and turkey aside.
Meanwhile, add the remaining 2 tablespoons of the oil to the skillet and set over medium heat. Crack the eggs into the pan and cook, covered, to the desired doneness, 2 to 4 minutes for sunny side up.
To make sandwiches, set four of the portobello caps upside-down on plates, top each with one-quarter of the tomato and turkey and one egg, then cover with the remaining caps.