Whisk 1¼ cups (156 g) all-purpose flour, ½ tsp. baking powder, and ½ tsp. Diamond Crystal or ¼ tsp. Morton kosher salt in a large bowl. Make a well in the center and add 1 large egg, 2 Tbsp. extra-virgin olive oil, and 2 Tbsp. plain whole-milk Greek yogurt. Blend with a fork, then gradually incorporate dry ingredients, mixing until combined and a shaggy dough forms. Knead dough in bowl to bring together, then transfer to a surface and knead until cohesive and smooth, about 3 minutes. Wrap dough in plastic and set aside at room temperature while you make the pilaf.
Filling and assembly
Place a rack in middle of oven; preheat to 400°. Rinse 1 cup basmati rice in a fine-mesh sieve under cold water until water runs clear; drain and transfer to a small bowl. Drain 2 lb. frozen chopped spinach, thawed, in sieve, pressing to expel as much liquid as possible. Transfer to a kitchen towel and squeeze over sink to expel any remaining liquid. Place spinach in a separate small bowl; reserve kitchen towel.
Heat 2 Tbsp. extra-virgin olive oil and 2 Tbsp. chilled unsalted butter in a large Dutch oven or other heavy pot over medium-high. Cook 2 medium leeks, white and pale green parts only, thinly sliced, stirring occasionally, until softened and becoming browned and crisp in spots, 5–7 minutes. Add rice and 2 tsp. Diamond Crystal or 1¼ tsp. Morton kosher salt and cook, stirring, until grains are translucent around edges, about 1 minute. Pour in 1¼ cups plus 2 Tbsp. water; stir to combine. Bring to a boil, then reduce heat to low. Cover with a lid and cook 12 minutes.
Remove pot from heat, uncover, and scatter spinach over rice. Cover pot with reserved kitchen towel, then with lid; let sit 5 minutes. Fluff rice gently with a fork and mix to distribute spinach. Stir in zest and juice of 1 lemon, 7 oz. feta, crumbled, ½ cup (packed) coarsely chopped dill, ⅓ cup golden raisins, ⅓ cup sliced almonds, and remaining 2 Tbsp. extra-virgin olive oil. Season with freshly ground pepper and more salt. Let cool slightly.
Roll out dough on a lightly floured surface to a 14″ round. (Dough will be very thin but quite elastic and malleable.) Brush bottom and sides of a 9″-diameter cake pan with some melted unsalted butter. Decorate bottom of pan as desired with dill sprigs, golden raisins, and sliced almonds. Transfer dough to pan. Lift edges and allow dough to slump down into pan, letting excess hang over sides; press dough into bottom and sides of pan.
Spoon filling into pan and spread into an even layer. Tugging dough gently (a bit like stretching pizza dough), fold edges up and over filling, overlapping slightly in the center (it’s okay if dough doesn’t quite cover filling; no one will see the gap once the pie is turned out onto a plate). Brush dough with more melted unsalted butter to help seal.
Cover pan with foil and crimp lightly around edges. Bake pie 20 minutes. Remove foil and continue to bake until crust is golden brown, 23–26 minutes more. Let cool 5 minutes, then invert pie onto a cake stand or large plate.
Combine 1 garlic clove, finely grated, and 1 cup plain whole-milk Greek yogurt in a small bowl. Thin with water to a pourable consistency; season with salt.
Cut pie into wedges with a serrated knife and serve with yogurt sauce for drizzling over.