1 lb plum tomatoes
cored and halved
1/2 medium white onion
cut into 1/2-inch slices
2 medium garlic cloves
peeled and smashed
1 medium serrano chili
1 tsp freshly squeezed lime juice
plus more as needed
1 tsp kosher salt
plus more as needed
4 medium poblano chilies (about 1 pound)
Freshly ground black pepper
3 cups shredded Monterey Jack cheese or queso fresco (about 8 ounces)
4 large eggs
separated and at room temperature
1/2 tsp kosher salt
plus more as needed
1 cup vegetable or canola oil
For the Sauce
Heat the broiler to high and arrange a rack in the upper third of the oven. Place the tomato halves skin-side up on a baking sheet. Scatter the onion, garlic, and serrano around the tomatoes. Broil until the tomato skins start to blacken and blister, about 10 minutes.
Transfer the vegetables to a blender, add the measured lime juice and salt, and blend into a smooth purée. Taste and season with additional salt and lime juice as needed. Transfer to a small saucepan and keep warm over low heat.
For the Peppers
Lay 1 chili on a work surface so that it sits flat naturally without rolling. Using a paring knife, make two cuts forming a “T” by first slicing down the middle of the chili lengthwise from stem to tip, then making a second cut perpendicular to the first about a 1/2 inch from the stem, slicing only halfway through the chili (be careful not to cut off the stem end completely). Carefully open the flaps to expose the interior of the chili and, using the paring knife, carefully cut out and remove the core. Scrape the inside with a small spoon to remove the seeds, ribs, and any remaining core. Repeat with the remaining peppers.
Turn 2 gas burners to medium-high heat. Place 1 chili directly on each burner and roast, turning occasionally with tongs, until blackened and blistered on all sides, about 5 to 7 minutes. (Alternatively, heat the broiler to high and arrange a rack in the upper third of the oven. Place all of the chilies directly on the rack. Broil, turning occasionally with tongs, until the chilies blacken and blister on all sides, about 8 to 10 minutes. The chilies will be softer using the broiler rather than a direct flame, so be careful not to tear them while stuffing.) Remove to a large heatproof bowl; repeat with the remaining 2 chilies.
Cover the bowl tightly with plastic wrap or a baking sheet and let the chilies steam until cool enough to handle, about 15 minutes. Using a butter knife, scrape away and discard the chili skins, being careful not to tear the chilies; set the chilies aside.
Heat the oven to 250°F and arrange a rack in the middle. Place a wire rack over a baking sheet and set aside.
Season the inside and outside of the chilies with salt and pepper. Stuff each chili, being careful not to tear them, with a quarter of the cheese (about a heaping 2/3 cup) and close the flaps over the cheese; set the chilies aside.
Whisk the egg yolks in a medium bowl until lightened in color and frothy, about 2 minutes; set aside.
Place the egg whites and measured salt in the clean, dry bowl of a stand mixer fitted with a whisk attachment. Beat on high speed until stiff peaks form, about 1 1/2 minutes. Remove the bowl from the mixer, add the egg yolks, and fold with a rubber spatula until just combined (do not deflate the egg whites); set aside.
Heat the oil in a large frying pan over medium-high heat until hot, about 4 minutes. Check to see if the oil is hot by submerging the handle of a wooden spoon or a wooden chopstick until it touches the bottom of the pan—the oil should bubble vigorously. Working with 1 chili at a time, drop 1/2 cup of the egg batter into the oil and use a rubber spatula to spread it to about the same size as the stuffed chili. Lay the chili seam-side down on top of the mound of batter. Drop another 1/2 cup of batter on top of the chili, spreading it with the rubber spatula to cover the sides and encase the chili. Cook without disturbing until the bottom of the chili relleno is golden brown, about 2 to 3 minutes. Using a flat spatula and a fork, carefully flip the chili relleno over and cook until the other side is golden brown, about 2 to 3 minutes. (If the sides of the chili are not brown, use a spatula or tongs to turn it onto each side to brown.) Transfer the chili relleno to the rack set over the baking sheet, season with salt, and place in the oven to keep warm. Repeat with the remaining stuffed chilies.
To Serve
Ladle about half a cup of the sauce onto each plate and place pepper on top of it. Garnish with sliced avocado and cilantro or scallions.