Toast almonds in a 4-qt. saucepan over medium-high, 4–5 minutes; let cool. Transfer to a food processor and pulse until finely ground; set aside. Add pigeons and stock to pan; boil. Reduce heat to medium-low; cook, covered, until pigeons are cooked through, 40–45 minutes. Using tongs, transfer pigeons to a cutting board; let cool, then shred meat, discarding skin and bones. Simmer cooking liquid over medium heat until reduced to 1 cup, about 30 minutes. Let cool and whisk in eggs; set sauce aside.

Wipe pan clean and add oil; heat over medium-high. Cook garlic and onion until golden, 8–10 minutes. Add ras el hanout, chile flakes, and saffron mixture; cook 1–2 minutes and remove from heat. Stir in reserved almonds, shredded meat, and sauce, the cilantro, parsley, salt, and pepper; set filling aside.

Heat oven to 400°. Grease a 9" springform pan with butter. Lay 1 sheet phyllo on a work surface and brush with melted butter. Fit into pan, allowing corners to hang over the edges. Repeat with another sheet phyllo. Spread ⅓ filling over dough. Repeat process of layering twice more. Fold corners of phyllo over filling. Top with remaining 2 sheets buttered phyllo; tuck corners around sides of pan, encasing the filling. Bake until golden and filling is set, 30–35 minutes. Let cool slightly, then unmold; dust with confectioners' sugar and cinnamon.