Forge the Creamy Bean-Yogurt Foundation : In a food processor, combine the rinsed cannellini beans, Greek yogurt, minced garlic, chopped dill, mint, and lemon zest. Blend until perfectly smooth and creamy. Transfer the mixture to a bowl, season generously with black pepper, and stir to combine. Set aside to allow the flavors to meld.
Construct the Verdant Base : Place a large skillet over medium heat. Add one tablespoon of the extra virgin olive oil and the sliced garlic, sautéing for 30 seconds until fragrant. Add the spinach and a tablespoon of water. Cover and cook for 1-2 minutes, just until the spinach has fully wilted. Squeeze the juice of half a lemon over the spinach, 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, steady simmer over medium heat. Crack one egg into a fine-mesh sieve set over a small bowl to drain away the thin whites, then transfer the egg to a small ramekin. Create a gentle vortex in the simmering water with a spoon and carefully slide the egg into the center. Cook for 3 minutes for a runny yolk.
Retrieve and Repeat : Using a slotted spoon, carefully lift the poached egg from the water and place it on a clean towel to drain. Repeat the poaching process for the remaining three eggs.
Infuse the Crimson Finishing Oil : While the final egg is poaching, combine the remaining 3 tablespoons of extra virgin olive oil and the Aleppo pepper in a small skillet over low heat. Warm the oil gently for 60-90 seconds, swirling the pan, until it becomes fragrant and takes on a deep red hue. Do not let it smoke. Remove from heat.
Assemble and Serve : Divide the wilted spinach evenly among four serving bowls. Spoon a generous amount of the herbed bean-yogurt mixture over the spinach. Top each bowl with a perfectly poached egg. Drizzle the warm, crimson-infused oil over the egg and yogurt. Garnish with a sprinkle of extra dill or mint and serve immediately.