Stir together the artichokes, yogurt, mint, 2 teaspoons of the olive oil and 1 teaspoon water in a small bowl. Set aside.
Heat a small saute pan over medium heat. Bring a small pot of water to a boil, then season with salt.
Add 1 tablespoon olive oil to the saute pan, then add the shallots and a pinch of salt. Cook until the shallots begin to caramelize, about 3 minutes. Break the eggs over the shallots, turn the heat down and cover the pan. Depending on how you like your eggs (my wife likes them runny), cook for about 3 minutes.
While the eggs cook, drop the spinach into the boiling water and blanch until wilted, using a slotted spoon or spider to mix it around to cook evenly. Drain the spinach, ring of excess water either in a colander or dish towel and then chop finely. Mix the spinach with some lemon juice and salt and a drizzle of olive oil.
Slide the eggs onto serving plates once they are cooked to your desired point (the heat will carry over, so removing them early never hurt anyone). Dress the whites of the eggs with the spinach, using your fingers to spread and crumble it. Drizzle with the artichoke sauce.