In a jar, combine the oil with the vinegar, olive paste, onion, basil and a generous pinch each of salt and pepper. Cover tightly and shake the vinaigrette until it is emulsified. Alternatively, whisk the ingredients in a medium bowl.
Halve, pit and peel the avocados. Thinly slice them crosswise, keeping them intact. Using a spatula, transfer the avocado halves, cut side down, to a platter and gently press each of them to fan out the slices. Drizzle the vinaigrette on top of the avocados and serve.