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Light the barbecue – you want the coals to be glowing hot. Meanwhile, if you’re making your own béarnaise, put the butter in a small pan over a very low heat and leave to melt. Skim off any scum from the surface and pour the clear (clarified) butter into a bowl, leaving behind the milky white solids that settle at the bottom.
Tip the tarragon, shallots, vinegar, 1 tbsp water and ½ tsp black pepper into another small pan and boil rapidly until the liquid has reduced to 1 tbsp. Put the egg yolks and 1 tbsp water in a bowl set over a pan of simmering water, ensuring the bowl isn’t touching the water, and whisk vigorously until voluminous and creamy. Remove the bowl from the pan and gradually whisk in the clarified butter until you have a thick, creamy sauce. Stir in the tarragon and shallot reduction and season with salt. Scrape into a serving bowl and keep warm in the oven on a low heat.
Season the ribs well on both sides. Barbecue for 9-10 mins on each side for rare (it should read about 55C on a meat thermometer) or 11-13 mins each side for medium-rare (60-65C). Remove and leave to rest for 10 mins. To serve, cut off the bone and carve the meat into long, thin slices. Serve with the béarnaise sauce, potatoes, carrots and runner beans, if you like (see below).