Step 1 Wash, dry and trim the fennel stalks till they meet the top and sides of the bulb. Remove dry or pulpy outer leaves, stalks and edible leaves. Discard or save the outer leaves and stalks for soup and the leaves for garnish.
Step 2 Slice the bulbs thinly crosswise. Place the slices in a large glass or non-reactive bowl with the onion, garlic, lemon and thyme. Mix in vinegar and sugar water. Season with 1 1/2 teaspoons salt and several grinds of pepper and cover with extra-virgin olive oil (or half olive oil, half vegetable oil), making sure that the olive oil covers the fennel.
Step 3Submerge the vegetables using a dish smaller than the circumference of the bowl, weighted with a kettle half-filled with water. Set aside for several hours at room temperature or, if making the day before, place in a tightly closed jar and refrigerate. This makes about 3 quarts marinated fennel, more than is needed for the remainder of the recipe. The salad will keep, covered and refrigerated, for 3 to 5 days; drain the salad before serving.