Melt the butter in a large frying pan over medium heat.
Add the onion, celery, thyme, bay leaf, measured salt, and measured pepper. Cook, stirring occasionally, until the onions are softened (about 5 minutes).
Add the flour and cook, stirring constantly, until the flour taste is cooked out, about 1 minute.
Add the half-and-half and milk, stir to combine, and simmer until it thickens slightly (about 5 minutes).
Add the turkey, walnuts, and brandy, stir to combine, and bring back to a simmer.
Continue to simmer until the alcohol flavor cooks out, about 2 to 3 minutes more.
Remove and discard the bay leaf. Taste and season with salt and pepper as needed.
Remove from heat and keep warm.
Poach the eggs in a pot of simmering, salted water in which a little vinegar has been added.
Serve the poached eggs over the hash, top with shredded cheddar cheese and enjoy!