Preheat the oven to 350°. In a food processor, combine the jalapeños, garlic, paprika, butter and 1 tablespoon of kosher salt and pulse until smooth. Season with salt and pepper. Transfer 1/4 cup of the jalapeño 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 jalapeño 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 jalapeño 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, adding 1 1/2 cups of water to the roasting pan and tenting the turkey with foil halfway through roasting. The turkey is done when an instant-read thermometer inserted deep in the thigh registers 170°. Uncover the turkey and transfer to a carving board; 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 sides of the pan. Strain the stock into the saucepan. Add the pan juices and bring to a simmer. Whisk in the reserved jalapeño butter with flour and simmer over moderate heat, whisking constantly. Cook until thickened, about 4 minutes. Season with salt. Carve the turkey and serve, passing the gravy at the table.