Preheat the oven to 200°C / 400°F. Use an ovenproof 10–12" (25–30 cm) skillet (cast iron preferred) so the ragù cooks evenly and you can transfer directly to the oven; preheating the oven ensures eggs set quickly and evenly without overcooking.

Toast and prep the pitas: place the 4 pitas on a rimmed baking sheet. Brush each pita with about 1/2 tsp of the reserved olive oil and sprinkle 1/4 tsp za'atar on each; press lightly so the za'atar adheres. Toast in the preheated oven for 6–8 minutes until crisp and golden at the edges. Doing this first means the pitas will be piping hot and crisp when the eggs are ready — and the za'atar's oils will bloom in the oven for more aroma.

Warm a large ovenproof skillet over medium heat and add the 4 tbsp extra‑virgin olive oil. Heat until the oil shimmers but does not smoke — this helps the aromatics release flavor without burning.

Add the chopped shallots and diced red pepper to the skillet and sauté gently for 3–4 minutes until the shallots are translucent and the pepper has softened. Sweating (not browning) the aromatics at this stage extracts sweetness and builds a mellow base for the ragù.

Stir in the minced garlic and cook 30–45 seconds until fragrant. Add the smoked paprika and red pepper flakes (if using) and toast for 15–20 seconds — toasting spices briefly in oil 'blooms' them, releasing volatile aromatic oils and deepening flavor.

Add the halved cherry tomatoes and 4 tsp tomato paste. Stir to combine and increase heat to medium‑high. The tomato paste adds concentrated umami; cooking it until it darkens slightly (1–2 minutes) caramelizes the sugars and deepens the ragù’s flavor.

Simmer the tomatoes, stirring occasionally, for 8–10 minutes. Smash larger cherry tomato halves with a wooden spoon as they soften so they break down into a chunky sauce. The goal is a glossy, slightly thickened ragù so excess water evaporates — if the ragù is too watery the eggs will poach rather than bake and you'll lose texture.

Season the ragù with a generous pinch of sea salt and freshly cracked black pepper, then fold in the chopped basil and about two‑thirds (about 80 g) of the crumbled feta. Stir just enough to integrate — the feta will add salt and creamy pockets when baked, so avoid over‑salting.

Using the back of a spoon, make four shallow wells in the ragù and crack two eggs into each well (or crack into a bowl and slide into wells for neatness). Space the eggs so they have room to set; this arrangement encourages even cooking and an attractive presentation.

Scatter the remaining feta over the ragù and eggs and transfer the skillet to the preheated oven. Bake for 8–10 minutes for whites set and yolks runny; 10–12 minutes for yolks more set. Baking rather than stovetop finishing gives gentle, even heat that cooks whites without making yolks chalky.

Remove the skillet from the oven and let rest 1–2 minutes (carryover heat will finish the eggs). Taste and adjust seasoning with a little more salt or black pepper if needed, and squeeze each portion with a lemon wedge to brighten the dish — acidity awakens the tomato and feta flavors.

Serve family‑style: slide each serving onto a plate or serve directly from the skillet with one crisp za'atar pita per person. Garnish with extra basil leaves and an extra pinch of smoked paprika or red pepper flakes if you like more heat.