Purée tomatoes, onion, and garlic with water and 3/4 tsp salt in a blender until smooth. Strain through a medium-mesh sieve into a bowl, pressing on and then discarding solids.
Heat 1/4 cup oil in a 12-inch heavy skillet over medium heat until it shimmers. Add tomato mixture (it will spatter) and simmer over low heat, stirring occasionally, until thickened slightly and a pool of oil forms around bubbles, about 20 minutes. Stir in enough of boiling-hot water to reach the consistency of a thin broth. Season with salt. Keep warm over low heat.
While sauce simmers, slit 1 side of each chile lengthwise and remove seeds and veins, keeping chiles intact. Fill each chile first with about 3 Tbsp packed Oaxaca cheese and then with about 2 Tbsp queso añejo and close chiles around filling as well as possible.
Put flour in a wide bowl and carefully dredge chiles in flour (make sure filling doesn't fall out), gently knocking off excess, and reserve on a plate.
Heat remaining cup oil in a 12-inch heavy nonstick skillet over medium heat until it shimmers.
Meanwhile, beat egg whites with a pinch of salt using an electric mixer at medium-high speed until they just hold soft peaks. Beat yolks briefly to loosen, then fold into whites with a rubber spatula gently but thoroughly.
Working in batches of 3, dip floured chiles, one by one, in egg batter to coat, transferring to skillet. Fry chiles, turning once with a slotted spoon, until batter is golden all over, about 1 minute. Transfer chiles as fried to warm tomato broth, turning to coat, then let stand 2 minutes so they absorb some of broth. Serve chiles with broth.
Note: Look for chiles that are soft and pliable enough not to need soaking.
Chiles can be stuffed and coated with flour 1 day ahead and chilled.
Tomato broth can be made 1 day ahead and chilled (covered once cool). Bring to a bare simmer before frying chiles.