For the roast chicken: Preheat the oven to 450 degrees F.
Sprinkle the cavities of the chickens with the salt and pepper; stuff each with 2 lemon halves, 2 garlic cloves and 2 thyme sprigs. Rub the skins with the clarified butter. Truss the chickens.
Place the chickens, breast side up, on a rack set into a roasting pan. Roast until golden brown, 10 to 15 minutes. Reduce the heat to 350 degrees and scatter the onions, carrots and celery around the chickens. Roast until the chickens are 160 degrees at the thigh, 45 to 60 minutes more. Remove to a cutting board and let rest for 10 to 15 minutes.
Place the roasting pan on the stovetop and cook until the vegetables are caramelized. Pour off all but 2 ounces of the chicken fat. Add the flour and cook for 2 minutes, whisking. Whisk the stock into the roux until smooth, then simmer for 20 minutes. Season the pan gravy with salt and pepper. Before serving, whisk the unsalted butter into the gravy.
Cut the chickens in half and remove the backbone.
For the sweet potato puree: Preheat the oven to 350 degrees F.
Pierce the skins of the sweet potatoes with a fork. Roast until tender, about 45 minutes.
Scrape the sweet potato flesh into a food processor. Add the butter and brown sugar, and puree until smooth. Season with salt and white pepper.
For the green beans: Bring 1 gallon water to a boil, and season with 2 ounces kosher salt. Set up an ice bath for shocking the green beans.
Add one-third of the green beans to the boiling water; cook until al dente, 2 to 3 minutes. With a slotted spoon, transfer the beans to the ice bath to stop cooking. Remove the beans from the ice bath and dry well. Repeat two more times, until all the green beans are blanched.
Heat 2 tablespoons grapeseed oil in a large skillet. Add the blanched green beans and saute for 1 to 2 minutes. Add the butter and the almonds, and cook for another 1 to 2 minutes. Season with salt and pepper.
To plate: Divide the sweet potato puree between 6 plates, and make a smear with the puree. Place one chicken half on each plate. Lean some of the green beans up against the chicken, and then spoon 3 to 4 ounces of the pan gravy over the chicken. Serve immediately.