This section is from the book "Cooking For Profit", by Jessup Whitehead. Also available from Amazon: Cooking for Profit.
A delicate piquant sauce to be served with roast fillet of beef, with the small, marinated steaks called by the French tournedos, with a simple grilled steak of small dimensions, with roast fowl or fish, is the sort of warm mayonnaise called by the French Bearnaise.
In the first place, get some fine butter, and set it to melt over a gentle fire. When the butter is just tepid, beat into it, with a fork, yolks of eggs; add aromatic herbs, finely-chopped, a dash of garlic, and a spoonful of good vinegar or lemon-juice, turning regularly with a wooden spoon until the mixture is of the consistency of a mayonnaise.
Mlle. Francoise's Bearnaise Sauce. - Put in a stewpan a dozen shalots, a seasoning bouquet, a little muscade, and a teaspoonful of freshly ground pepper, the whole moistened with a glassful of vinegar. Boil down, and then strain through a sieve. Now take a small saucepan, and put in it a big lump of butter of the best quality, three yolks of very fresh eggs; add two tablespoonfuls of the liquid already prepared as above, and put the whole over a very gentle fire; turn it briskly with a wooden spoon, until the sauce gets thick, and take it off the fire very sharply, before it turns oily.
 
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