Preheat the broiler in your oven. Place the poblano peppers on a baking sheet and broil, turning occasionally, until the skins are charred and blistered on all sides, about 8-10 minutes.
Once the peppers have steamed, carefully peel off the charred skin, taking care not to tear the peppers.
Stuff each pepper with strips of Monterey Jack cheese, filling them as full as you'd like.
In a shallow bowl, beat the egg whites until stiff peaks form. In another bowl, beat the egg yolks until smooth and creamy. Season with salt and pepper.
Heat vegetable oil in a large skillet or deep fryer to 350°F (175°C). Spread the flour on a plate. Roll each stuffed pepper in flour until coated, shaking off any excess.
Heat the tomato sauce or enchilada sauce in a saucepan over medium heat. Garnish with chopped cilantro, sour cream, and sliced avocado, if desired.