Preheat the oven to 425ºF (220ºC). Line a 12-cup muffin tin with paper liners; spray the liners and the pan with non-stick cooking spray. Note: this recipe makes 16 muffins, so bake one batch first, keep the remaining batter covered, and then bake the remaining batter.
Whisk the flour, baking powder, baking soda, and salt together in a large bowl until combined.
In a mixing bowl, combine melted butter, oil, sugar, buttermilk, vinegar, and vanilla.
Fold the wet ingredients into the dry ingredients and mix everything together by hand. Do not overmix; the batter will be thick and a little lumpy.
Toss the strawberries with 2 teaspoons of flour (this will keep them from sinking to the bottom). Set 1/2 cup of the strawberries aside. Add the remaining berries to the batter and fold with a spatula until evenly distributed. Do not overmix.
Fill the muffin cups right to the top. Scatter the reserved strawberries evenly over the muffins, then sprinkle the turbinado sugar on top of each muffin, if desired.
Bake for 5 minutes, then reduce the oven temperature to 350ºF (180ºC) and continue baking until golden brown and a toothpick inserted in the center comes out clean, another 20 to 23 minutes.
Allow the muffins to cool for 10 minutes in the muffin pan, then transfer to a wire rack to continue cooling.