2 tbsp. olive oil
1 large or 2 medium onions, chopped
A sprinkle of sugar
1 ½ cups chicken stock
1 tbsp. plus 1 1/2 tsp. butter
1 tbsp. plus 1 1/2 tsp. flour
2 tbsp. white wine
½ cup whole milk
Black and white pepper
About 1 1/2 tsp. Worcestershire sauce
½ tsp. granulated garlic
High-heat cooking oil (such as canola or grapeseed)
1 lb. boneless rib eye steak, very thinly sliced
1 cup shredded sharp provolone
4 sub or hoagie rolls (each 10 to 12 inches long)
Chopped fresh flat-leaf parsley, for serving
In a small to medium skillet, heat the oil, two turns of the pan, over medium. Add the onion; season with salt and a sprinkle of sugar. Cook, stirring occasionally, until the onion softens, 12 to 15 minutes. Add 1 cup stock. Raise heat to medium-high and cook, stirring often, until the liquid absorbs into the onion, about 5 minutes.
Heat a large cast-iron griddle or skillet over high.
In a small saucepan with a lid, melt the butter over medium to medium-high heat. Whisk in the flour. Cook, whisking often, until the flour is absorbed, about 1 minute. Add the wine, then the remaining 1/2 cup stock. Cook, stirring often, until the sauce thickens a bit, about 1 minute. Whisk in the milk and bring to a simmer; season with salt, black pepper, and white pepper. Whisk in the Worcestershire and granulated garlic. Reduce heat to low and cover to keep the gravy hot.
Fill a squeeze bottle with high-heat oil (or hold a finger over the opening of the oil bottle to slow the pour). Drizzle the griddle with some oil. Scatter a few slices of steak on the griddle; season. Working in batches, cook, turning and chopping the steak with the side of a metal spatula, until crisp and brown, 2 to 3 minutes per batch. Transfer the steak to a plate.
Add the cheese to the gravy and stir in a figure eight motion until the cheese melts, about 3 minutes. Remove from heat. Pile the steak onto the rolls. Top with the onion, cheese gravy, and parsley.