Special equipment: a deep-frying thermometer
1. Combine 1/4 cup of the olive oil, the soy sauce, lime juice, orange juice, 1 teaspoon salt and 1/4 teaspoon black pepper in a large resealable plastic bag. Add the chicken pieces and seal the bag, making sure the marinade is distributed evenly in the bag and coating all of the chicken pieces. Marinate for 3 1/2 hours in the refrigerator, then 30 minutes at room temperature before cooking.
2. Put the remaining 1/4 cup olive oil, the garlic and jalapenos in a small saucepan over medium heat. When the garlic starts to brown at the edges, remove from the heat and add the cilantro, vinegar and 1/2 teaspoon salt. Set aside to cool.
3. Whisk together the flour, paprika, oregano, 1 teaspoon salt and 1/2 teaspoon black pepper in a medium bowl. Remove the chicken pieces from the marinade and pat dry. Dredge the pieces in the seasoned flour and place on a wire rack.
4. Heat 3 inches of vegetable oil in a large heavy-bottomed pan until a deep-frying thermometer inserted in the oil registers 300 degrees F. Add half of the chicken pieces, making sure not to overcrowd the pan. Cook until lightly browned, about 5 minutes. Remove the chicken with a spider or slotted spoon to a paper-towel-lined baking sheet. Repeat with the second batch of chicken.
5. Heat the oil until the temperature reaches 375 degrees F. Fry the chicken pieces a second time, until the exteriors are very crisp and dark brown, about 2 minutes for the wings and breasts and 4 minutes for the thighs and drumsticks. Remove to another paper-towel-lined baking sheet. Serve with the sauce on the side.