Char the poblano chiles directly over a gas flame or under a broiler, turning, until the skins are blistered and blackened in most spots, 8–12 minutes.
Transfer to a bowl, cover, and let steam 10 minutes. Peel off the skins, make a 2-inch slit along one side of each chile, and carefully remove seeds and membranes without tearing.
Stuff each chile with Oaxaca cheese and gently press the slit closed.
Pat the chiles dry, then dredge lightly in all-purpose flour, shaking off any excess; set on a tray.
For the sauce, blend the roma tomatoes, white onion, garlic cloves, vegetable broth, Mexican oregano, and kosher salt until smooth. Pour into a saucepan with the bay leaf, bring to a gentle simmer over medium heat, and cook until slightly thickened and the raw tomato taste has mellowed, 10–12 minutes; keep warm.
For the batter, place the large eggs with whites in a clean bowl and beat until stiff peaks form, 2–3 minutes; beat the yolks briefly, then gently fold them into the whites just until incorporated.
Heat the vegetable oil in a wide skillet to a depth of about 0.5 inch to 350°F, or until a drop of batter sizzles immediately on contact.
Working in batches, hold a floured chile by the stem, coat it in the egg batter, and lay it in the hot pan; fry until puffed and deep golden, 2–3 minutes per side. Drain on a rack or paper towels.
Spoon some warm sauce onto each plate and set a chile on top, or simmer the fried chiles in the sauce for 2–3 minutes to coat. Serve immediately.