Heat the olive oil in a large, deep skillet over medium heat. Add the onions, carrots and celery. Cook the vegetables for 3-4 minutes, stirring frequently.
Add the garlic. Cook 1-2 minutes, stirring almost constantly.
Raise the heat to high and the ground turkey, tomato paste, salt, oregano, cumin and black pepper and immediately start to stir and break up the meat using a wooden spoon or a spatula.
Cook for 3-4 minutes to brown the turkey while stirring and continuing to break it up.
When the turkey is browned, carefully add the white wine to the skillet; cook about 1-2 minutes until most of the liquid evaporates.
Lower the heat to medium-low and add the tomato sauce, water and olives to the skillet. Stir well to combine.
Cover the skillet and cook for 20 minutes to allow the flavors to develop and come together. Stir occasionally and keep it at a simmer. If the liquid is boiling too vigorously, lower the heat a bit.
If the picadillo is too saucy, uncover the skillet and let some of the liquid cook out. Stir often.
Taste the picadillo and add salt if needed. As a reference, we did not add extra salt to ours.
Let the meat cool some, then place it in a bowl and refrigerate until completely cooled.