Preheat the oven to 350°F (175°C). Line a baking sheet with parchment paper.
Arrange the wonton halves on the prepared sheet in a single layer. Lightly brush each with olive oil if desired.
Bake the wonton triangles until golden and crisp, about 9 minutes. Remove and let cool on the sheet.
Whisk together the soy sauce, rice vinegar, lemon juice, wasabi paste, and toasted sesame oil in a mixing bowl until smooth.
Add the cubed tuna, avocado, and cucumber to the dressing. Gently fold in chopped chives to taste, being careful not to mash the avocado.
Portion the tartare onto the cooled wonton crisps and sprinkle with sesame seeds and additional chives if desired before serving.