Preheat the oven to 425°F. Line two baking sheets with parchment paper.
Heat the vegetable oil and butter in a medium skillet over medium heat. Add the onions and cook for about 8 minutes, until very soft. Reduce the heat to low and cook for another 20 to 25 minutes, until the onions are golden and caramelized. Remove from the heat and season with salt and pepper, to taste. Transfer the onions to a small bowl and set aside.
Add the spinach to the same pan and cook over medium heat for 1 to 2 minutes, until wilted. Drain and set aside.
Place one sheet of puff pastry on a lightly floured work surface and roll out to a 12-inch square. Cut out five 3-inch circles and transfer to the prepared baking sheet. Repeat with the remaining sheet of puff pastry. Prick the dough all over with the tines of a fork, leaving a ¼-inch border around the edges. Divide the caramelized onions, spinach, and feta between the tartlets. Top each tartlet with a slice of bresaola.
Bake for about 20 minutes, until the crust is golden brown all over. Remove from the oven and serve warm or at room temperature.