CH [Christopher Hirsheimer] was given this version of the classic French red wine–reduction sauce by friends and owners of a West Coast restaurant she took her daughters to when they were little girls. The silky sauce was served over beautiful little steaks and over chopped steak for the girls. Now we always dress up our chopped steak with this elegant sauce.
For the sauce, put the shallots and wine in a heavy medium saucepan and cook over medium heat until the wine has been reduced to ¼ cup and the shallots have softened, about 10 minutes. Add the demi-glace (or bouillon paste dissolved in 2 cups of water), thyme, and bay leaf, and simmer until the sauce is reduced by half and thickens slightly, 10–20 minutes. Strain the sauce through a fine sieve into a bowl, discarding the solids. Return the sauce to the pan over low heat. Add the butter one piece at a time, whisking after each addition until it is completely incorporated before adding the next. Whisk until the sauce is smooth with a lovely sheen. Season to taste with salt and pepper. Stir in the parsley. Cover and set the pan in a larger pan filled with hot water to keep the sauce warm.
For the chopped steak, put the meat, shallots, parsley, thyme, egg yolk, and salt and pepper to taste in a medium bowl. Using two forks, gently mix everything together without compacting the meat. Form the meat into 4–6 oval patties. Use a knife to crosshatch the surface of the patties on both sides. Heat the oil in a heavy large nonstick or cast-iron skillet over medium-high heat. Put the patties in the skillet and cook on both sides until brown, about 3 minutes per side for medium-rare.
To serve, spoon a little warm sauce onto a plate, place a patty on top of the sauce, and spoon more sauce on top. Repeat with the remaining patties. Garnish with chopped parsley. Serve with fries, if you like.