In a pot, place the veal in cold water, bring to a boil, and skim off any foam. Reduce the heat and add the chopped carrots, onion with a clove, celery, thyme, and bay leaf. Season with salt and white pepper. Simmer slowly and gently for 1.5 to 2 hours until the meat is very tender.
Once the meat is cooked, remove it from the broth along with the vegetables and set them aside. Strain the broth through a sieve.
In a saucepan, melt the butter and add the flour to create a 'roux'. Cook for a few minutes. Gradually pour in 500 ml of the strained broth, stirring constantly until the sauce thickens. Add the heavy cream and let it simmer.
In a small bowl, whisk the egg yolks with a little lemon juice. Ladle some of the hot sauce into the egg yolk mixture and whisk slowly (tempering). Then, slowly pour this mixture back into the saucepan with the sauce, stirring constantly. Heat gently, but do not let it boil, or the egg yolks will curdle.
Add the cooked meat, carrots, and sliced mushrooms (you can sauté them briefly in butter beforehand) to the sauce. Stir to combine everything evenly. Season to taste as needed. Serve immediately, sprinkled with fresh parsley, preferably with rice or mashed potatoes.