Bring 3 inches of water to a boil in a large wide pot. Heat the vegetable oil in a large nonstick skillet over high heat until very hot. Season the steak with salt and pepper, add to the skillet and cook until browned, 2 to 4 minutes per side for medium rare. Remove to a cutting board to rest.
Reduce the heat under the skillet to medium; add the spinach and 2 tablespoons water and cook, stirring occasionally, until wilted, about 2 minutes. Season with salt and pepper. Divide the English muffin halves among plates and top with the spinach.
Meanwhile, combine the egg yolk, butter, lemon juice and 1 tablespoon water in a small saucepan over medium heat. Cook, whisking, until the butter melts and the sauce thickens, about 3 minutes. Add the tarragon and season with salt and pepper. Cover and keep warm.
Add the vinegar to the boiling water, then reduce the heat to maintain a gentle simmer. Crack an egg into a small bowl or ramekin and carefully slide the egg into the water; repeat with the remaining eggs. Cook until the whites are set but the yolks are still jiggly, about 3 minutes. Remove the eggs with a slotted spoon, blot the bottoms dry on a kitchen towel and place on the English muffins.
Slice the steak against the grain and divide among the plates. Spoon the sauce over the eggs.