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