Egg and Broccolini Sandwiches

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3 tablespoons extra-virgin olive oil

1 3/4 pounds broccolini (2 to 3 bunches), cut into 2-inch pieces

3 cloves garlic, finely chopped

Kosher salt and freshly ground pepper

6 large eggs

8 slices sharp provolone cheese

4 hoagie rolls, split

Sliced pepperoncini, plus brine from the jar, for topping

Potato chips, for serving (optional)