Rub a cast iron skillet or heavy pan with EVOO and heat over a medium-high flame. When the pan is very hot, add the steaks and cook for 4 minutes on the first side, then 3 minutes on the second side for medium doneness. The steaks should be deeply caramelized before turning.
Place a skillet on the stovetop for the eggs and heat over medium-low flame.
Meanwhile, to a small skillet over medium heat add a drizzle of EVOO and 2 tablespoons butter. Add the shallot and sprinkle in the thyme leaves and sauté until soft, 2-3 minutes. Sprinkle with flour and stir a minute. Add the wine and stir to evaporate, 30 seconds. Add the Worcestershire sauce, stock, lots of pepper and a pinch of salt. Stir in the mustard and incorporate. Then stir in the half-and-half or cream and stir for 1 minute more. Turn off the heat.
Toast the bread while the sauce cooks and butter it while it's hot.
Add 1/2 tablespoon of the remaining butter to the hot egg skillet over medium-low heat and cook the eggs to desired doneness, two eggs at a time. Repeat with the remaining butter and two eggs.
Serve the steaks on halved buttered toast with eggs on top. Pour the sauce over the steak and eggs and garnish with watercress.