In a sauce pan over medium heat, combine tarragon, shallot, white wine vinegar, white wine, salt and pepper until reduced to 1 tablespoon. This process is refered to as vinegar reduction. Set aside to cool.
Melt butter in the same sauce pan. Once melted, leave it at the stove to keep it hot while preparing the other ingredients.
In a blender, add the egg yolks, cooled vinegar reduction and salt. Use the sauce program and blend ingredients for 30 to 60 seconds.
While blender is running, slowly pour hot butter in the blender lid opening.
Towards end of blending cycle, add tarragon or mix it by hand.
In a cast iron or pan, add oil over medium high heat.
Fry depending on your preference rare, medium rare, medium well or well done.
After making Bearnaise, keep it in the microwave to keep warm while making steak.
Extra Bearnaise will keep well in the fridge in a seal container for about 3 to 4 days.
Use it for poach eggs for breakfast, fish or another steak dinner. Check my detailed Bearnaise post and video I linked above. I discussed how to warm leftover Bearnaise.
If you like your steak grilled, you may do so.
If you like rib eye roast instead of individual steaks, check out my 2 previous post. Link is listed above in blog.