Thaw the pastry. Thaw frozen phyllo fully according to its package directions before starting; thawing time is additional. Keep the package closed until the filling, brushing mixture, and pan are ready.
Dry and salt the spinach. Wash the spinach, dry thoroughly, remove tough stems, and cut into strips about ½–¾ inch (1–2 cm) wide. Toss with the filling salt and massage for about 1 minute. Let stand for 10 minutes, then squeeze firmly in small handfuls to remove released liquid.
Mix the filling. Mash the drained white cheese finely with a fork. Stir in the quark, the filling egg, and black pepper. Loosen the squeezed spinach and fold it through. The filling should be moist and spreadable without free liquid. Keep refrigerated while preparing the other components.
Whisk the brushing mixture. Whisk the yogurt with its separate egg until smooth. Gradually whisk in the brushing oil, then briefly mix in the sparkling water. Stir again regularly during layering so the oil stays distributed.
Heat and prepare the pan. Preheat a conventional oven to 400°F (200°C), a practical Fahrenheit rounding. Use a lower-middle rack. Thinly coat a metal baking pan about 15 × 11 inches (38 × 28 cm) with the pan oil.
Divide pastry and filling. Trim or arrange the sheets to cover the pan; usable offcuts can lie flat between intact sheets. Reserve roughly one-quarter of the pastry for the bottom and one-quarter for the top. Divide the remaining half into three similar intermediate bundles. Divide the filling into four equal portions. Keep waiting sheets under parchment topped with a barely damp towel, without the towel touching the pastry.
Build the bottom. Lay in the bottom bundle one sheet at a time, brushing each lightly with the yogurt mixture. Spread the first portion of filling evenly and loosely over it. Avoid pressing it into a dense layer.
Add the middle layers. Add the first intermediate bundle sheet by sheet, brushing each lightly, then spread the second filling portion. Repeat with the second bundle and third portion, then the third bundle and final filling portion. Keep the four filling layers thin and even.
Cover and brush the top. Add the reserved top sheets individually, brushing each lightly all the way to the edges. Distribute any remaining brushing mixture evenly across the surface rather than leaving a puddle in one place.
Bake until set. Bake about 35–40 minutes, until the top and edges are evenly golden and the bottom is baked. Check the central egg-and-cheese filling for at least 160°F (71°C). If the surface browns early, tent loosely with foil for the final baking stage and continue until the center and base are done.
Rest and portion. Let the baked pie stand uncovered for 5 minutes. Cover the pan with parchment and then a clean, dry kitchen towel for another 15 minutes. Divide into the number of servings shown and serve warm. Uncover after resting to avoid trapping excess steam.