In a heavy-bottom saucepan, melt butter. Whisk in flour, stirring constantly, about 2 min. Mixture will bubble. Do not let it brown. Whisk in hot milk, making sure there are no lumps and mixture is smooth. Continue to stir as sauce thickens. Bring to a boil. Add salt and pepper, lower heat, and cook, stirring for 2–3 min. Remove from heat. To cool this sauce for later use, cover it with plastic wrap. Set aside.
Peel onions and cut them in half, lengthwise. Cut into thin half-moon slices. Over medium heat, sauté onion with 4 tbs. olive oil and thyme sprigs until caramelized. Add stock and cook until most is evaporated. This may take about 20 min. Remove thyme sprigs. Salt and pepper. Set aside.
Heat a grill pan over high heat. Lightly brush pan with olive oil. Generously salt and pepper both sides of each burger. Place burger on grill pan and cook 5–6 min. each side for medium burger. After burger is flipped, place cheese on top and cook until desired internal temperature is reached and cheese is melted.
Split the English Muffins with a fork. Toast and butter each side. Then, layer on a tbs. of béchamel sauce, followed by meat and sautéed onions.
The French Onion Soup Burger was created by Chef Paul Denamiel, owner of Le Rivage and Little Prince in New York City