Pat the chicken dry with paper towels. Using kitchen shears, cut along both sides of the backbone and remove it, then flip the chicken breast-side up and press firmly to flatten (spatchcock).
In a bowl, whisk the grated garlic, ajĂ panca paste, dried oregano, ground cumin, black pepper, kosher salt, soy sauce, white vinegar, dark beer, vegetable oil, and lime juice until smooth. Reserve 2 tablespoons of this marinade in a small bowl for basting; refrigerate it separately.
Loosen the skin over the breasts and thighs with your fingers. Rub the remaining marinade under the skin and all over the chicken. Place the chicken on a rimmed sheet pan, cover, and refrigerate 2–4 hours.
Set a rack in the middle of the oven and preheat to 425°F (220°C). Fit a wire rack inside a rimmed sheet pan and lightly oil the rack.
Transfer the chicken, skin side up, to the prepared rack. Roast 20 minutes.
Baste the chicken with the reserved marinade, rotate the pan, and continue roasting 25–35 minutes more, until the thickest part of the thigh registers 165°F/74°C and the skin is deep mahogany.
For extra char, switch to broil and cook 1–3 minutes, watching closely, until small patches blister and darken.
Transfer the chicken to a board and rest 10–15 minutes before carving.
Meanwhile, make the green sauce: In a blender, combine the cilantro, jalapeño, huacatay paste, ajà amarillo paste, mayonnaise, lime juice, chopped garlic, vegetable oil, kosher salt, and black pepper. Blend until smooth and spoonable, 20–40 seconds, scraping down as needed. Chill until serving.
Carve the chicken into breasts, thighs, drumsticks, and wings. Serve with the green sauce and lime wedges.