Forge the Creamy Bean–Yogurt Foundation: In a food processor, combine the cannellini beans, Greek yogurt, minced garlic, dill, mint, and lemon zest; blend until smooth. Transfer to a bowl, season with black pepper, and stir in the remaining lemon juice.
Construct the Verdant Base: Warm 1 tablespoon olive oil in a large skillet over medium heat. Add the sliced garlic; sauté 30 seconds. Add spinach and 1 tablespoon water; cover and cook 1–2 minutes until wilted. Sprinkle with 1 tablespoon lemon juice, season with black pepper, and toss. Remove from heat.
Master the Silken Poach: Fill a medium saucepan with at least 4 inches of water and bring to a gentle simmer. Crack an egg into a fine-mesh sieve over a small bowl to drain thin whites, then into a ramekin. Create a gentle vortex and slide the egg in. Poach 3 minutes for a runny yolk (4–5 for medium). Poach up to 2 eggs at a time.
Retrieve and Repeat: Lift eggs with a slotted spoon and drain on a clean towel; repeat for remaining eggs.
Infuse the Crimson Finishing Oil: While the final egg cooks, warm the remaining 3 tablespoons olive oil with Aleppo pepper in a small skillet over low heat for 60–90 seconds until fragrant and deep red; do not let it smoke.
Assemble and Serve: Divide spinach among four shallow bowls. Spoon the herbed bean–yogurt over the spinach, top each with a poached egg, and drizzle with the warm Aleppo oil. Garnish with extra dill or mint and serve immediately.