How to Make It
In a medium bowl, whisk the 1/4 cup of olive oil with the lemon juice, salt and sugar. Divide the marinade between 2 large, resealable plastic bags. Add the lemon slices, herb sprigs and garlic to the bags. Add the halved hens and turn to coat well. Seal the bags and let the hens stand at room temperature for 45 minutes.
Roast the jalapeño directly over a gas flame or under the broiler, turning, until charred all over; let cool. Stem and seed the jalapeño and transfer it to a food processor. Add all of the remaining ingredients except the oil, salt and white pepper. Pulse the mixture until very finely chopped, then add the 1 cup of olive oil and pulse until blended. Season the salsa with salt and white pepper and transfer to a bowl.
Light a grill. Remove the hens from the marinade and scrape off the herbs, garlic and lemons. Pat the hens dry, brush with olive oil and season with salt and white pepper. Oil the grill grates and grill the hens skin side down over moderate heat, turning occasionally, until lightly charred in spots and an instant-read thermometer inserted in the thighs registers 160°, 25 to 30 minutes. Transfer the hens to platters and let rest for 10 minutes before serving with the salsa verde.