Place sweet potatoes, olive oil, 1 teaspoon curry powder, and a dash of salt in a plastic bag. Shake until well coated. Spread on a cookie sheet. Bake, until golden and cooked through, stirring once at the halfway point, 18 to 20 minutes.
While the sweet potatoes are cooking, combine tomatoes, chickpeas, carrots, red pepper flakes, and remaining curry powder in a medium saucepot. Bring to a boil and immediately reduce to a simmer. Add cooked sweet potatoes, stir in the spinach, and cook until spinach is wilted, 1 to 2 minutes
Divide among the pita, top each sandwich with 1/4 of the cilantro leaves and 1/4 of the yogurt. Serve with lime wedges.