To make the pasta: Sift together the flour and salt and pour onto a clean cutting board or work surface. Make a well in the center of the flour mound.
In a small bowl, whisk together the olive oil and the wine. Add a little of the olive oil wine mixture into the center of the well at a time, begin slowly working the liquid into the flour, making sure to maintain an outside wall of flour until the very end. Keep working by hand until all of the liquid has been incorporated and you have created a homogenous ball of dough. If the dough is too dry add 1 teaspoon of water at a time until the dough comes together. Continue kneading the dough for 5 minutes more. Set the dough aside to rest, covered with a damp cloth, for 30 minutes.
Preheat the oven to 350 degrees F.
Carefully slice off the tips of the artichokes, remove all of the tough outer leaves, and peel the base. Cut each artichoke in half and remove the fuzzy choke in the middle. Slice each half into 1/4-inch slices.
In a 12-inch saute pan, heat 1 tablespoons olive oil over a medium-high flame until the oil is hot but not smoking. Add the garlic and parsley and cook until the garlic is soft and light golden brown. Add the lamb and cook, stirring occasionally, until the meat is brown on all sides, about 8 minutes. Add the artichokes and cook another 5 minutes, until carmelized. Add the white wine and let simmer for a few minutes. Season with salt, to taste.
Using the remaining 1 tablespoon of olive oil, lightly grease a 9-inch round baking pan with 2-inch sides.
In a small bowl, mix together the egg, yogurt, and 1 tablespoon chopped parsley. Season with salt and pepper, to taste, then set aside.
Using a lightly floured rolling pan, roll out the dough until it forms a 9-inch circle. Gently lay the dough at the bottom of the pan. Spread the lamb and artichoke mixture over the dough. Top with an even layer of hardboiled egg slices, then with the egg and yogurt mixture. Season with a sprinkle of chili flakes, black pepper, a drizzle of extra-virgin olive oil and grated Pecorino.
Place in the oven and bake until the yogurt is golden brown, about 40 minutes.