With a mandolin or a sharp knife, carefully cut the potatoes into ¹⁄₁₆-inch (1 mm) thick slices.
Add the potato slices to a large bowl and cover with water. Toss the potatoes around to remove excess starch, then drain.
Add 1 teaspoon of salt, the pepper, paprika, and garlic powder and toss the potatoes until evenly coated.
Add the oil to a small bowl. Dip a paper towel in the oil and grease the cups of a 12-cup muffin tin.
Arrange 5 potatoes around the sides of a muffin cup, overlapping slightly. Place 1 slice on the bottom to create a flower. Repeat with the remaining potato slices. Sprinkle shredded cheese over the potato cups.
Bake for 10 minutes.
Remove the potato cups from the oven and reduce the oven temperature to 300°F (150°C).
Sprinkle bacon over the cheese. Add an egg to each cup. Sprinkle with salt.
Bake for another 20-25 minutes, or until the egg whites are mostly cooked.