Step 1Cut off the fennel stalks and slice the bulbs widthwise into thin strips. Mix the stalks and slices in a bowl with the onion, garlic, lemon and thyme. Mix in the vinegar and sugar-water, if desired.
Step 2Season with salt and pepper, transfer to a 1-quart jar, and cover with olive oil. Make sure that the olive oil covers the fennel. (Use the bottom of a bowl or other weight to press the vegetables down under the olive oil.) Wait several hours before serving. Store in a closed jar in the refrigerator. More fennel may be added to the same marinade throughout the week.