With the rack in the middle position, preheat the oven to 400°F (200°C). Line a baking sheet with parchment paper.
In a large skillet, soften the zucchini in 2 tbsp (30 ml) of the oil. Season with salt and pepper. Add the spinach and olives and cook for about 2 minutes, stirring constantly. Add the pesto and stir well. Drain the vegetables in a sieve and set aside to cool. Add to the tartare.
On a work surface, brush a sheet of phyllo dough with some of the remaining oil. Cover with a second sheet and brush with more oil.
Cut the dough lengthwise into three strips. Place 2 tbsp of the tartare mixture at the end of each strip of dough. Fold the dough over the filling to form a small triangle. Fold the triangle on itself several times to form a small turnover. Repeat with the remaining ingredients.
Place the turnovers on the baking sheet and brush with oil. Bake for about 20 minutes or until the dough is golden brown.