1.To make béarnaise sauce, place vinegar, peppercorns, onion, the water and half of the tarragon in a small saucepan. Bring to the boil. Reduce heat and simmer, uncovered, until liquid is reduced to 2 tablespoons. Strain liquid into small jug and discard solids. Place egg yolks in small heatproof bowl over small saucepan of simmering water; do not allow water to touch base of bowl. Whisk in reserved liquid; add butter in a thin, steady stream, whisking constantly until mixture thickens slightly. Stir in remaining tarragon.
2.Cut bacon into 2cm x 14cm strips.
3.Spread a little pesto on one side of each bacon strip; wrap one strip, pesto-side against the beef, around each piece of beef. Secure with a toothpick.
4.In batches, cook filet mignons, on a heated oiled grill plate (or grill or barbecue) until browned both sides and cooked as desired.
5.Serve hot with béarnaise sauce.
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