Make a bouquet garni by placing the celery, thyme, bay leaf, and leek white together. Tie with twine to form a bundle. Make the beurre manie by placing 2 tablespoons of softened butter in a small bowl with the flour and mix the 2 together with your fingers until smooth. Break the mixture into pea-sized pieces and reserve.

Melt the remaining 2 tablespoons of butter in a saucepot. Add the onions and cook to color lightly. Then add the wine, crushed garlic, and bouquet garni. Bring to a boil, lower the heat immediately, and simmer uncovered for 20 minutes. Lift the onions and bouquet garni out with a slotted spoon and discard.

Strain the wine through a sieve and return it back to the pan. Heat until simmering. Break an egg into a teacup, and then slide it gently into the hot wine. You can poach up to 4 eggs at 1 time. Poach the eggs for 4 minutes, them lift them out with a slotted spoon. Place them on a plate and cover with aluminum foil to keep warm while poaching the second batch of eggs. Place these eggs on a plate and cover with foil as well. Add the beurre manie to the wine and bring it to a boil, stirring constantly. Season the sauce with salt and pepper.

Rub 1 side of each piece of toast with a piece of garlic. Place the toast into the bottom of shallow bowls. Place 2 eggs on top of each piece of toast. Spoon the Beaujolais sauce over the eggs and serve.