Peel, halve and cut potatoes into ¼-inch thick slices. Toss together sliced potatoes, salt, pepper and garlic powder together.
Heat a 9-inch non-stick saute pan over medium heat. Add oil and as soon as the oil gets moderately hot, add potatoes, stir well. Cover pan with a lid, cook (stirring every 2-3 minutes), until potatoes are tender, about 9-12 minutes (a knife inserted into one or two pieces should go in without any resistance). Don't worry if any of the potatoes break apart while stirring. It's all going to be covered any way. Take pan off heat and continue making sandwiches
Make sandwich
Preheat a waffle iron.
Spread some mayo one side of each bread slices. Place 4 slices (mayo side down) on a plate. Spread 2 tablespoons basil pesto on each slice, top with ¼th cooked potatoes, a slice of cheese and top with remaining bread (mayo side up).
Cook sandwiches, one at a time, close waffle iron and cook until deep golden and crisp, about 3-4 minutes. Repeat with remaining sandwiches, serve immediately (if not, refer notes)
Notes
If not eating immediately, then place cooked sandwiches on a wire rack to cool. This helps in not making the bottoms soggy.Sandwiches can be cooked in a skillet instead of a waffle iron, just as any grilled cheese - Wipe the same skillet clean with a paper towel, place a sandwich, cook until the bottom is golden (pressing sandwich with a flat spatula, this helps in the sandwiches cooking faster/ crispier ) gently turn sandwich over, cook until the other side is golden as well (pressing again).