Set strainer lined with double layer of cheesecloth over medium bowl. Place yogurt in strainer; cover and chill 3 to 5 hours to drain. Transfer yogurt to small bowl. Stir in mint; season with salt and pepper. (Can be made 1 day ahead. Chill.)
Place 1/2 cup warm water and saffron in small bowl; let stand at least 20 minutes to infuse.
Sprinkle lamb with salt and pepper. Heat 2 tablespoons oil in heavy large pot over medium-high heat. Working in batches, cook lamb until brown on all sides, adding more oil as needed, about 5 minutes per batch. Transfer lamb to large bowl. Pour all but 1 tablespoon fat from pot (or add 1 tablespoon oil if dry); heat pot over medium heat. Add onions; sprinkle with salt and pepper. Sauté until beginning to brown, about 5 minutes. Add lemon, garlic, ginger, cinnamon, and cayenne. Stir 1 minute. Add saffron mixture; stir, scraping up browned bits. Add tomatoes with juice, figs, and lamb with any juices to pot. Stir to coat. Add 2 1/2 cups broth.
Bring stew to boil. Reduce heat to medium-low, then cover with lid slightly ajar and simmer until meat is tender, stirring occasionally and adding more broth by 1/4 cupfuls as needed if dry, about 1 1/2 hours. Season to taste with salt and pepper. (Can be made 2 days ahead. Cool slightly. Refrigerate uncovered until cold, then cover and chill.)
Bring stew to simmer, thinning with more chicken broth if necessary. Divide stew among 6 plates; top each serving with dollop of minted yogurt. Place wedge of Crispy Noodle Cake alongside each and serve.
