Preheat a large skillet over medium heat. Add olive oil and onions and let cook for 4 to 6 minutes, stirring every 2 minutes.

Add shaved ribeye meat to the skillet. Stir in garlic and season with salt and pepper. Continue to cook until steak is fully cooked, about 10 minutes.

Transfer meat mixture to a large mixing bowl.

Place an egg roll wrapper on a clear work surface like a diamond. Add a slice of cheese on top of the filling. Wet corners of the wrapper with beaten eggs.

Fold bottom corner over the filling, fold 2 sides inwards, and roll up tightly.

Heat oil in a large Dutch oven to 350°F (180°C). Check temperature with a deep fry thermometer.

Working in batches, fry 2 to 3 egg rolls at a time in the hot oil, making sure not to overcrowd. Fry until golden brown, about 2 to 5 minutes per side.

Transfer fried egg rolls to a paper towel-lined plate to drain.

Meanwhile, stir together ketchup and mayo for the dipping sauce.