Position an oven rack 6 inches from the broiler. Preheat the broiler.

Line a rimmed baking sheet with aluminum foil. Line a plate with paper towels.

Arrange the peppers in one layer on the prepared baking sheet and broil a few inches from the heat until blackened on each side, 4 to 6 minutes per side. Transfer the roasted peppers to a medium heat-proof bowl. Cover the bowl with plastic wrap and let the peppers rest for 15 minutes. Use paper towels to gently rub the blackened skin off the peppers. Keeping the stems intact, cut a 3-inch slit down one side of each pepper, then gently remove the seeds using paper towels.

In a medium bowl, mix together the cheddar and pepper jack cheeses. Fill each pepper, through the slit, with half the cheese until full but able to be held shut with your fingers. Set aside.

Gently fold the egg yolk mixture into the egg whites, being careful not to deflate the egg whites; you want a fluffy air-bubbly batter that can coat the back of a spoon. Don’t overmix or your batter will be too thin.

In a small (8-inch) skillet, add enough olive oil to come 1/2-inch up the sides, about 1/2 cup. Heat over medium-high heat until shimmering hot, then reduce the heat to medium.

Working quickly, dip one stuffed pepper into the batter, pinching the pepper’s opening shut. Spoon more batter over the top of the pepper, covering all but the stem. Return the heat to medium-high and remove the pepper from the batter, wipe away any excess, then carefully lay the coated pepper down in the hot oil. When the side in the oil is golden and flaky-crisp, after about 1 minute, use tongs to gently turn the pepper and fry the other side until similarly golden, about 1 minute per side; to fry the battered pepper near the stem, you may need to use tongs to hold it at an angle. Transfer the finished pepper to the prepared plate and sprinkle with salt. Scoop out and discard leftover bits from the oil then repeat with the remaining pepper.