Preheat the oven to 350 degrees. Spread the pecans in a pie plate and toast in the oven for about 10 minutes. Transfer the pecans to a food processor and let cool completely.

In a medium skillet, heat the oil. Add the garlic and cook over moderate heat until very lightly golden, about 1 minute. Add the sage leaves and fry, stirring gently, until crisp, about 2 minutes. Using a slotted spoon, transfer the sage leaves and garlic clove to a paper towel-lined plate and let cool. Add half of the sage leaves and the garlic clove to the food processor along with the butter and 1 tablespoon of salt; pulse until smooth. Transfer 1/4 cup of the butter to a small bowl and stir in the flour; reserve.

Beginning at the neck end, gently separate the turkey skin from the breast and legs using your fingers. Season the turkey cavity with salt. Rub half of the pecan-sage butter from the food processor under the skin, spreading it over the breast and thighs.

Set the turkey on a rack in a large roasting pan and scatter the carrot and onion in the pan; add 1 1/2 cups of water. Rub the remaining pecan-sage butter from the food processor all over the outside of the turkey. Roast on the bottom rack of the oven for 3 1/2 to 4 hours, until an instant-read thermometer inserted deep in the thigh registers 170 degrees; halfway through roasting, add 1 1/2 cups of water to the roasting pan and tent the turkey with foil. Transfer the turkey to a carving board and let rest for 30 minutes.

Meanwhile, strain the pan juices into a large measuring cup; discard the solids. Spoon off the fat and discard it. (You should have about 2 cups of defatted pan juices.) In a large saucepan, boil the turkey stock until it is reduced to 3 cups, about 15 minutes. Set the roasting pan over 2 burners on high heat; add the reduced stock and scrape up any browned bits stuck to the bottom and side of the pan. Strain the stock into the saucepan, add the pan juices, and bring to a simmer. Whisk in the reserved pecan-sage butter with flour and simmer over moderate heat, whisking constantly, until thickened, about 4 minutes. Season with salt.

Carve the turkey, transfer to a platter and garnish with the reserved sage leaves. Serve the turkey, passing the gravy at the table.