PARBOIL 3 CLOVES OF GARLIC FOR ABOUT A MINUTE.
POACH EGG IN WATER WITH A SPLASH OF VINEGAR UNTIL THE YOLK IS BARELY SET. ABOUT 3 MINUTES. REMOVE WITH A SLOTTED SPOON AND COOL UNDER GENTLY RUNNING WATER.
SQUEEZE ONE CLOVE OF RAW GARLIC THROUGH A GARLIC PRESS INTO A FOOD PROCESSOR. ADD A PINCH OF SEA SALT, ESPELETTE PEPPER, 2 TBSPS OF DIJON MUSTARD, 1 TSP LEMON JUICE, 1 TSP WATER. PULSE SEVERAL TIMES TO DISSOLVE MUSTARD AND MIX THE INGREDIENTS.
ADD THE EGG YOLKS AND THE POACHED EGG. TURN ON THE FOOD PROCESSOR AND SLOWLY ADD THE OILS IN A THIN STREAM. GO SLOWLY IN THE BEGINNING, PAUSING TO MAKE SURE THE OIL IS EMULSIFIED BEFORE ADDING MORE.