Trim enough of the root ends of the leeks to dispose of the roots yet still keep the leek halves held together at the root ends. Trim the tops so that only the white and some of the light green tops remain. Discard the dark green tops or reserve for another use.
Place the trimmed, halved leeks in a medium saucepan and add the chicken stock. It should be enough to cover the leeks by 1-inch or so. If not, add water as needed to cover the leeks. Add 1 teaspoon of the salt and bring to a gentle boil, weighting the leeks with a small heatproof plate or other heatproof object in order to keep the leeks submerged while cooking. Reduce heat to a simmer and continue to cook the leeks until they are very tender when pierced with the tip of a knife, 12 to 15 minutes. Using tongs or a slotted spoon, carefully remove the leeks from the poaching liquid and set aside on a paper-towel lined plate to cool. Once cooled, cover the leeks with plastic wrap and transfer to the refrigerator until thoroughly chilled, 1 to 2 hours. (Note: You can strain the leek poaching liquid and reserve for another use, such as for soup or risotto.)
While the leeks are chilling, make the vinaigrette. In a medium non-reactive bowl combine the remaining 3/4 teaspoon salt, vinegar, shallots, mustard, and pepper and whisk to combine. Let it sit for 5 to 10 minutes. Combine the vegetable oil and olive oil in a measuring cup. While whisking constantly, slowly drizzle the oil into the vinegar-shallot mixture until the vinaigrette is smooth and thick. Whisk in the parsley and chives. Taste and adjust seasoning, if necessary. Refrigerate until ready to serve.
Preheat the oven to 400 degrees F.
In a small bowl combine onions, butter and thyme and mix well. Loosen the skin from the chicken and stuff some of the onion butter underneath each breast. Season each chicken breast with salt, pepper, and Essence.
Heat an ovenproof saute pan over high heat. Add the oil and heat. Place the chicken breasts in the pan, skin side down, in batches as necessary, and saute until skin is golden brown, about 5 minutes. Turn breasts over, place in the oven and cook until chicken is cooked through, about 10 minutes.
When ready to serve, divide the leeks among 6 plates, crossing the 2 pieces in an X-fashion, and drizzle generously with the vinaigrette. Garnish with the halved grape tomatoes on the side. Top the leeks with the chicken, drizzle with additional vinaigrette if desired, and serve immediately.
Combine all ingredients thoroughly.
Recipe from "New New Orleans Cooking", by Emeril Lagasse and Jessie Tirsch, published by William and Morrow, 1993.