For dressing, in a screw-top jar combine oil, vinegar, shallot, wasabi or horseradish, sugar, fish sauce (if desired), sea salt, and pepper. Cover and shake well; set aside.
Halve cherries, if desired; set aside 1/4 cup of the cherries. In a large salad bowl combine the greens, remaining cherries, and the avocado. Toss with dressing. Top salad with almonds, basil, and reserved 1/4 cup cherries.
Mash cherries. Place in a clean, heatproof jar. In a small stainless steel saucepan heat vinegar to boiling. Pour over cherries in jar. Cool slightly; cover with a nonmetallic lid. Let stand at room temperature 3 to 5 days. Strain out cherries; cover and store vinegar in a cool place up to 3 months. Makes 4 to 6 servings.