Preheat the oven to 400 degrees F.
In a large saucepan, over medium heat, add the oil. When the oil is hot, add the veal, brown for 2 minutes. Add the onions, carrots, and celery. Season with salt and pepper. Saute for 2 minutes.
Add the garlic, tomatoes, peppercorns, and stock. Bring the liquid to a boil. Reduce the heat to medium-low, cover and simmer for 30 minutes, or until very tender. Remove from the heat and cool completely. Strain the veal, reserving the liquid.
In a food processor, fitted with a metal blade, add the veal and puree until smooth. Add the butter, bread crumbs, and 1/2 cup of the reserved poaching liquid. Season with salt and pepper.
Using a sharp knife, split the back of the quail down the middle. Season the quail with Essence.
Divide the stuffing into eighths. Form the mixture into round balls. Place each quail, skin side down, and place each ball of veal mixture in the center of the quail. Wrap the quail around the stuffing. Place each quail on a parchment lined baking sheet, breast side up.
Brush each quail with the melted butter. Place the quail in the oven and roast for 20 to 25 minutes. In a saucepan, over medium heat, add the Madeira. Bring the liquid to a boil and reduce by half, about 4 minutes. Add the reduction and simmer for 2 to 3 minutes.
In a saucepan, over medium heat, add the milk and remaining butter. Season with salt and pepper. Bring the liquid to a boil. Whisk in the grits. Stir constantly for 6 minutes or until tender. Remove from the heat and stir in the cheese. Season with salt and pepper. Remove from the heat and keep warm.
To serve, spoon the grits in the center of each plate. Place two quails on top of each plate of grits. Spoon the sauce over the quail.
Combine all ingredients thoroughly and store in an airtight jar or container.
Recipe from "New New Orleans Cooking", by Emeril Lagasse and Jessie Tirsch. Published by William and Morrow, 1993.