small glass bowls or large ramekins, for broth to dip burgers
Halve and peel the onions and leave root end attached. Thinly slice the onions from the root end, down and in opposite direction, thinly slicing onion, then slice off the root end to separate.
To a medium nonstick skillet over medium to medium-high heat, add the oil, 1 turn of the pan. Melt butter into oil, then add onions and season with salt and white or black ground pepper, bay leaf. Stir and soften a few minutes, then add 1/2 cup water and about 1/2 cup consomme or beef bone broth and let it absorb, 4 to 5 minutes. Add creme fraiche or sour cream and mustard, then combine and reduce heat to low/warm, cover. Remove bay before serving.
Warm the remaining 2 1/2 cups consomme or bone stock in small pot over low heat for dipping.
Preheat cast-iron skillet or griddle for burgers over medium-high heat.
Meanwhile, place potatoes in a skillet with lid with 2 1/2 cups water and some butter and bring to boil, covered. Cook 8 to 10 minutes to just tender, then remove lid and let water evaporate. Crush potatoes a bit with the bottom of a water glass and brown to crisp up edges and skins, 6 to 8 minutes. Season with salt and pepper.
While the potatoes are cooking, mix garlic into the beef. Score meat into 4 sections, then form 4 patties thinner at centers than edges for even cooking. Season the meat with salt and pepper. Cook patties in skillet or on griddle about 8 minutes, turning occasionally. Melt cheese, 2 slices per patty, over tops in last minute or so.
Split and lightly toast buns, optional.
Drain and slice some cornichons for the buns.
To serve: Top bun bottoms with pickles, patties, creamy onions, tops of buns. Cut burgers in half and serve with small bowls of warm consomme or stock for dipping burgers and potatoes alongside, sprinkled with chopped parsley.