How to Make It
Place beef cheeks on a wire rack set inside a rimmed baking sheet; season with thyme leaves and 1 tablespoon black pepper. Place thyme sprigs on top of beef cheeks. Cover and chill 8 hours or overnight. Let beef cheeks stand at room temperature 1 hour and 30 minutes before cooking.
Sprinkle beef cheeks with 1 tablespoon salt. Preheat oven to 325°F. Heat 3 tablespoons oil in a large Dutch oven until oil is fragrant and almost smoking. Add beef cheeks, and sear, turning occasionally, until deeply browned on all sides, 15 to 20 minutes; transfer to a plate. Reduce heat to medium-high; add onion, carrot, and celery, and cook, stirring occasionally, until vegetables are lightly caramelized, about 5 minutes. Stir in red wine, port, and vinegar, scraping bottom of Dutch oven to loosen any browned bits. Bring to a boil over high, and cook, stirring occasionally, until reduced by three-fourths, about 15 minutes. Add stock, and bring to a boil.
Add parsley, bay leaves, and beef cheeks and their juices. Cover Dutch oven tightly with aluminum foil; cover with lid. Braise in preheated oven until meat is tender and easily pierced with a paring knife, about 3 hours. Remove from oven, uncover, and set aside. Increase oven temperature to 400°F.
Let meat rest 30 minutes in juices, then transfer meat to a large plate. Pour braising liquid through a fine wire-mesh strainer over a bowl; discard solids. Let strained liquid stand 5 minutes; skim off and discard fat. Return beef cheeks and strained braising liquid to Dutch oven.
Roast, uncovered, at 400°F until meat is hot and caramelized on top, about 15 minutes.
Toss together chicories, lemon juice, and remaining 1 tablespoon oil. Season with remaining 1/8 teaspoon black pepper and remaining 1/4 teaspoon salt.
To serve, divide hot green-garlic soubise evenly among 6 large bowls; top evenly with dressed chicories. Place beef cheeks on chicories, and ladle some braising liquid over each serving.