For the Thai grilled chicken: Spatchcock each chicken by placing it breast-side down on a cutting board with the legs closest to you. Use kitchen shears to cut down the backbone. Flip the chicken over and press down in the center of the breast to flatten. Prick the skin side all over with a fork.
Add the cilantro roots, fish sauce, palm sugar, peppercorns, oyster sauce, turmeric, salt and 4 cloves of the garlic to a food processor. Process until all the peppercorns are broken up and a chunky paste is formed.
Rub the chicken all over with the paste (including under the skin) and marinate at least 8 hours and up to 24 hours.
Meanwhile, cook the remaining 8 cloves garlic with the oil in a small saucepan over medium heat, stirring constantly, until light golden brown, about 3 minutes. Strain through a fine mesh sieve, capturing the oil in a small bowl beneath. Spread the garlic on a plate to cool; reserve for serving.
Before grilling, bring the chicken to room temperature for an hour. Spoon 2 tablespoons of the garlic oil over the chicken.
Preheat the oven to 350 degrees F. Heat a double-burner grill pan over medium heat.
Place chickens skin-side down on the grill pan at a 45 degree angle. Cook until marked by the grill pan, flipping once to create crosshatches, about 5 minutes. Transfer skin-side up to a baking sheet fitted with a wire rack.
Trim off the spiky top and the base from the lemongrass, peel off the first few outer layers and then bruise the stalk with the heel of a chef's knife; this helps release the flavorful oils. Using the bruised lemongrass as a brush, baste the chickens all over with some of the remaining garlic oil.
Cook in the oven until the chickens are golden brown and an instant-read thermometer inserted in thickest part registers 165 degrees F, 10 to 15 minutes. For the last 2 minutes of baking, switch the oven to broil.
Let the chicken rest for 10 minutes and then cut each into pieces. Garnish with the reserved garlic.
For the Thai green papaya salad: Add the garlic, chiles, fish sauce, palm sugar and lime juice to a food processor. Process until a chunky paste is formed. (At this point you can adjust the flavor to your liking; add more fish sauce for a saltier taste, more lime juice to make the salad more sour.)
Transfer the paste to a large bowl and add the papaya, long beans, tomatoes and the dried shrimp and 1 tablespoon of the peanuts if using. Mix together by hand, rubbing the paste into the salad, making sure to coat everything.
Transfer the salad to a serving dish, making sure to get all the dressing. Top with the remaining 1 tablespoon peanuts if using.
The salad also goes really well with fried chicken and sticky rice.