Preheat oven to 400 degrees. Clean mushrooms and roast until fragrant, set aside.Reduce oven temperature to 350.
Preheat large dutch oven or stock pot on stove over medium high heat. Add 3 tablespoons olive oil. Season short ribs with salt and pepper then add to pan and sear on all sides until browned. Remove short ribs from pan, set aside. Add thyme and stir for about one minute, Do not burn.
Add onion, carrot, celery and thyme to the pot and season well with salt and pepper. Cook for about 5 minutes, add the garlic and cook another 5 minutes. Pour in the red wine and beef stock, bring to a boil.
Place the short ribs into the vegetable and stock mixture making sure to "nestle" them into the liquid. The braising mixture should not completely cover the ribs (only about 2/3 of the way). If there are too many ribs to fit into a single layer in the pan, divide the vegetable mixture and use two pots. Do not stack the ribs on top of each other.
Place the lid on the pot, leaving it "ajar" so steam may escape. Place pot in the oven and cook until tender but still in one piece (not falling apart); about 2 1/2 to 3 hours.
While the ribs are cooking, prepare the compound butter by roughly chopping the mushrooms then mincing the mushrooms in a food processor or finely by hand. Stir the cooled mushrooms into the softened butter. Spoon the butter on to a sheet of plastic wrap and roll into a cylinder with the diameter a little larger than a quarter. Chill until ready to use.
While short ribs are cooking, prepare the Pear and Celery Salad.
To make the salad, cut 3 pears and 3 celery stalks into 2 1/2 inch long pieces then cut into 1/4 by 1/4 inch sticks. Be sure to keep some of the celery leaves on the small, thin stalks at the top of the celery stalk and use them in the salad. Keep in chilled water to stay crisp.
Mix the creme fraiche with the juice of 1/2 lemon. Season with salt and pepper.
When the ribs are almost finished cooking. Remove about 2 cups of the liquid and reduce slightly in a sauce pan until thickened.
Remove short ribs from pan and place on warm serving plater. Top with a slice of mushroom compound butter. Cover with foil to keep warm.
Remove the sauce from the heat and strain though a fine sieve, whisk in cubes of butter one at a time. Spoon the sauce over the short rib and butter
Plate the short ribs on individual plates or on a serving plater. Spoon the sauce over the short ribs and butter.
Toss the pears and celery alumettes (sticks) in the cream fraiche and serve on top of each short rib.
Trim the leaves and tender parts of the flat leaf parsley and sprinkle on the top of the salad.