Preheat: set oven to broil for 5 minutes.
Line a baking sheet with foil or a silicone baking mat or foil.
Roast the Peppers: Lay the peppers on the baking sheet and place under broiler for about 5-7 minutes until the skins blacken. Carefully turn the peppers and continue to cook the other side for an additional 5-7 minutes until the skin blackens.
Place the roasted peppers in a plastic bag or tupperware and seal so that the steam is captured. Allow these to sit for about 10 minutes before moving onto the next step.
Remove Skins: Using a paring knife, scrape the skin off of the roasted peppers. Get as much of the blackened skin off as possible.
Slice Peppers: Carefully slice into the peppers to open them, just a small slit in the middle of the pepper, leave the tops and only slice enough to open them, don’t slice them completely in half.
Remove Seeds: Scrape out the membranes and seeds if desired. Poblanos aren't very spicy and this step can be skipped.
Stuff the peppers with cheese and fold the peppers closed again. Use toothpicks if needed to close them. (Just remember they are there so you remove them!!)
Prepare for Frying: Pour the oil to a large dutch oven and heat over high heat until the oil is hot enough that a drop of batter will sizzle when dropped in. Prepare a large baking sheet lined with paper towels or a cooling wire rack.
Prepare the batter: divide the eggs placing the egg yolks into a small bowl and the egg whites into a mixing bowl. Beat the egg whites until stiff peaks form.
Add in the flour, salt and the egg yolks and mix until combined. Batter will be light, smooth and fluffy.
Coat the Peppers: Dip the chiles into the batter and coat both sides. Gently place the peppers into the hot oil. Be careful not to splatter.
Fry the Peppers: Allow the chiles to fry for about 4-5 minutes, until the batter is a dark golden-brown color, then carefully flip and continue frying the other side an additional 3-5 minutes. Depeding on the size of your pan you can fry 1-2 chiles at a time, be careful not to overcrowd the pan.
Remove from the oil and set onto the prepared baking sheet. Remove the toothpicks after cooling for a couple minutes.