Egg and Potato Sandwiches

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2 tablespoons plus 2 teaspoons extra-virgin olive oil

1 medium russet potato, peeled and cut into 1/2-inch cubes

1 large Italian frying pepper (Cubanelle), seeded and thinly sliced

1 medium onion, thinly sliced

Kosher salt

6 large eggs

4 cups baby arugula

2 teaspoons red wine vinegar

4 seeded semolina rolls, split and toasted

4 slices sharp provolone cheese (optional)