Step 1

Combine cumin, paprika, oregano, 1 tablespoon salt, and 1/2 teaspoon pepper. With kitchen shears, cut chicken along its backbone along one side; open like a book. Place on a rimmed baking sheet, breast-side up; press down flat. Rub all over with 2 tablespoons oil and grated garlic. Sprinkle with spice mixture. Let stand 1 hour (or refrigerate, covered, overnight).

Step 2

Preheat oven to 425 degrees. In a saucepan, cover potatoes with 2 inches of water and bring to a boil; add a generous amount of salt. Boil until just tender, 11 to 13 minutes. Drain and immediately return to pan; toss with remaining 3 tablespoons oil and season with salt and pepper. (Potatoes will break apart slightly.)

Step 3

Scatter potatoes around chicken.  Roast until chicken is golden and a thermometer inserted into thickest part of breast registers 160 degrees, 45 to 50 minutes. Transfer chicken to a plate. Flip potatoes and continue to roast until golden and crisp, 5 to 10 minutes more.

Step 4

Meanwhile, purée cilantro, jalapeño, chopped garlic, mayonnaise, lime juice, and 1 tablespoon water in a blender until smooth. (If sauce is too thick, add more water, 1 teaspoon at a time.) Season with salt.

Step 5

Carve chicken and arrange on platter with potatoes. Sprinkle with cilantro leaves and serve with cilantro sauce and lime wedges on the side.