Stuffed Sweet Potatoes with Curried Chickpeas and Mushrooms

Preheat oven to 450°F. Toss chickpeas, mushrooms, curry powder, cumin, 2 Tbsp. oil, and 3/4 tsp. salt on a rimmed baking sheet. Divide mushrooms and chickpeas into separate piles and spread in a single layer.
Cut potatoes in half lengthwise. Rub with 1 Tbsp. oil and 1/2 tsp. salt and transfer cut side down to another rimmed baking sheet.
Roast vegetables and potatoes, rotating pans halfway through and arranging spinach over vegetables during the last 2 minutes of cooking, until chickpeas are golden brown and crisp and potatoes are fork-tender, about 25 minutes.
Meanwhile, juice limes to yield 6 Tbsp. juice. Blend yogurt, 4 Tbsp. lime juice, and 1/2 cup chopped cilantro in a blender until smooth.
Transfer spinach, mushrooms, and chickpeas to separate bowls. Transfer potatoes to a serving platter. Using the back of a spoon, gently push down in center of potato halves to create a divot.
Spoon vegetables into divots, then top with yogurt sauce and cilantro leaves.
Potatoes, chickpeas, and mushrooms can be cooked 3 days ahead; store separately, cover, and chill. Reheat in oven before serving.