Roast Chiles: Roast chiles 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 to 6 minutes. (Or broil on rack of a broiler pan about 2 inches from heat.) Immediately transfer to a large bowl and cover, then let stand 20 minutes.
Make tomato sauce while chiles stand: Puree all sauce ingredients except oil with 1 1/4 teaspoons salt in a blender until smooth, then strain through a medium-mesh sieve into a bowl, pressing on solids, then discarding them.
Heat the 2 tablespoons oil in a deep 12 inch heavy skillet or heat proof casserole (preferably one that can go to the table) 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 salt.
Stuff and fry chiles: Carefully rub off skins from chiles, leaving stems attached. Poblano chiles: 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). For Anaheim chiles: slice off the top 1/2 inch of stem end including seed placenta and remove any loose seeds. Wipe chiles clean with a paper towel if necessary.
Stuff chiles with cheese and enclose filling by overlapping slit slightly if possible, then transfer to a plate. (Use a toothpick to mark any hotter chiles if desired).
Heat 1/2 inch oil in another 12 inch heavy skillet over medium heat until it registers 360°F on thermometer.
While oil heats, beat egg whites with 1/8 teaspoons salt using an electric mixer until they just hold stiff peaks. Whisk yolks with 1/8 teaspoons salt in a large bowl, then gently fold in whites.
Put flour in a fine-mesh sieve and dust chiles, turning them to lightly coat all sides.
When oil is ready,coat chiles, 1 at a time (hold chile by stem or use spoons) in egg batter, spooning it over chile to cover completely; then fry, turning once, until golden on all sides (lap oil over any lighter areas), 4 to 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 or ladle sauce over individual servings. (makes extra sauce).