For the eggs and sauce: Bring the wine and stock to a slow boil in a saute pan. Gently crack the eggs in and poach until the whites are set and the yolks runny, 2 to 3 minutes. Remove to a plate.
Remove any clumps of egg white and bring the wine-stock mixture back to a boil. Add the bay leaf, carrots, celery, garlic, onions and thyme. Simmer until reduced by half and concentrated, about 20 minutes.
For the garnish: While the sauce reduces, melt 1 spoonful of the butter in a frying pan and cook the mushrooms, about 3 minutes. Remove. Add the second spoonful of butter and fry the bacon until brown. Remove the bacon with a slotted spoon and drain on paper towels. Fry the onions in the fat left in the pan until tender and golden, about 10 minutes. Drain off the excess fat and put the bacon and mushrooms back in.
Once the sauce has reduced, knead the butter and flour together to form a paste. Whisk it, a piece at a time, into the simmering sauce until thick enough to coat a spoon. Strain the sauce over the garnish. Bring to a boil, and check the seasonings. Gently place the poached eggs in the sauce to reheat for a minute. Serve the eggs in wide, shallow bowls with some garnish and sauce. Scatter with the parsley.