Chiles Rellenos (Cheese-Stuffed Poblano Peppers)

Roast 4 large fresh poblano chiles (about 1 lb.) on their sides on racks of gas burners (1 or 2 per burner) over medium-high heat, turning with tongs, until skins are blistered and slightly charred, 4–6 minutes. (Or broil on rack of a broiler pan about 2" from heat.) Immediately transfer to a large bowl and cover; let stand 20 minutes.

Purée 1½ lb. tomatoes, coarsely chopped, ¼ cup chopped white onion, 2 large garlic cloves, 1 Tbsp. distilled white vinegar, 1 tsp. sugar, 1 tsp. dried oregano (preferably Mexican), 1¼ tsp. Diamond Crystal or ¾ tsp. Morton kosher salt, and 1 cup water in a blender until smooth. Strain through a medium-mesh sieve into a bowl, pressing on solids; discard solids.

Heat 2 Tbsp. vegetable oil in a deep 12" heavy nonreactive skillet or ovenproof casserole over medium-high heat until it shimmers. Carefully pour in sauce and simmer gently, stirring occasionally, 10 minutes (sauce should still be thin). Season with kosher salt. Do Ahead: Sauce can be made 3 days ahead. Chill, uncovered, until cool, then cover.

Carefully rub off skins from chiles, leaving stems attached. Cut a slit lengthwise in each chile with kitchen shears, then carefully cut out seedpod (including attached ribs) without tearing chile (hotter chiles smell stronger when cut). Wipe chiles clean with a paper towel if necessary.

Divide ½ lb. Monterey Jack cheese, coarsely grated (2½ cups), among chiles and enclose filling by overlapping slit slightly if possible; transfer stuffed chiles to a plate. (Use a toothpick to mark any hotter chiles if desired.)

Fill a 12" heavy frying pan with ½" of vegetable oil and heat over medium until oil registers 360°F on thermometer.

Meanwhile, beat 4 large egg whites with ⅛ tsp. kosher salt using an electric mixer until they just hold stiff peaks. Whisk remaining 4 large egg yolks with ⅛ tsp. kosher salt in a large bowl, then gently fold in whites.

Put ¼ cup all-purpose flour in a fine-mesh sieve and dust chiles, turning them to lightly coat all sides.

When oil is ready, dip chiles, one at a time (hold chile by stem), in egg mixture, spooning it over chile to cover completely, then fry, turning once, until golden brown on all sides (lap hot oil over any lighter areas), 4–5 minutes. Transfer chiles with a slotted spoon to paper towels to drain.

Meanwhile, reheat sauce over low heat. Lay chiles in sauce to serve. Do Ahead: Chiles can be roasted and stuffed 1 day ahead. Chill, covered in an airtight container.