Preheat oven to 450°. On a rimmed baking sheet, toss peppers, red onion (separating pieces), and 3 tbsp. oil to coat. Sprinkle with 1 tbsp. zaatar. Roast vegetables, stirring once, until tender and blackened at edges, 30 minutes. Set aside. Reduce oven to 375° and set a rack on bottom rung.
Meanwhile, cook lamb and 1 tbsp. zaatar with remaining 1 tbsp. oil in a large frying pan over medium-high heat, breaking into pieces, until brown, 5 to 6 minutes. With a slotted spoon, transfer meat to a bowl. Sauté yellow onion and 1 tsp. zaatar in pan 3 minutes. Reduce heat to medium and cook until onion is golden, 8 to 10 minutes more. Stir in lamb.
Unroll filo on a work surface and trim to 13 in. square; save scraps for other uses. Move filo stack to the side and drape with a damp towel. Lay 1 sheet flat in center of work surface and lightly brush with butter, being sure to get corners. Lay a second sheet on top, offsetting corners like you're moving a hand 2 hours forward on a clock; brush with butter. Repeat, rotating each sheet to use all of filo and all but 1 tbsp. butter. Gently ease stack into a 10-in. pie pan (preferably glass), snugly fitting bottom into corners and letting edges overhang. Drape with damp towel.
Beat cream cheese in a bowl with a mixer until very smooth. Stir in feta. Drop spoonfuls into pie pan and gently spread to cover bottom of filo. Evenly arrange lamb, then pepper mixture on top.
Gently fold edges of filo over filling flush with rim of pan, loosely pleating, to partially cover peppers. Re-melt remaining 1 tbsp. butter if necessary, brush over filo rim, and sprinkle with remaining 3/4 tsp. zaatar.
Bake on bottom rack until filo is crisp and deep golden, about 40 minutes (in a glass dish, underside should look golden). Let cool at least 15 minutes. Eat warm or at room temperature, topped with cilantro sprigs, cut into wedges, and served with green sauce.
Vegetarian version: Omit lamb. In step 1, use 6 peppers (plus red onion), 1/4 cup oil, and 2 tbsp. zaatar. In step 2, cook yellow onion and zaatar in 1 tbsp. oil.
Make ahead: Let cool completely, then chill, covered, up to 1 day. Bring to room temperature (about 2 hours) before serving.