For the steak, preheat a large saute pan over medium-high heat. Oil the rump steak and then place into the preheated pan. Cook for about 2 minutes on the first side, and then flip and finish cooking the second side, about 2 minutes more. This will yield a steak that is on the rare side; cook 1 to 2 minutes longer, per side, for a more well-done piece.
Let rest for 5 to 10 minutes.
For the Sauce Bearnaise, put the shallot, tarragon stalks, 1 tablespoon of the chopped tarragon and the chervil, if using, and the vinegar, wine, and peppercorns in a heavy-bottomed saucepan and boil until reduced to about 1 tablespoon. Don't move from the stove; this doesn't take long.
Press the reduced liquid through a regular or tea strainer and leave to cool. Put egg yolks and water in the top of a double boiler. Set over the bottom, in which water has come to a simmer. Add the reduced and strained liquid and whisk well. Keep whisking as you add the butter, cube by cube, until it is all absorbed. Taste, season as you wish with the salt and pepper, and add lemon juice as you wish. Stir in the remaining tablespoon of fresh chopped tarragon as you're about to serve with the steak.