In a medium mixing bowl, stir together the flour, salt, baking powder, and baking soda. Set aside.
Using an electric mixer fitted with the paddle attachment (or hand mixer), mix together the softened butter, granulated sugar, and brown sugar until combined, about 2 minutes. Stop the mixer and scrape down the sides of the bowl.
With the mixer on low, add the eggs one at a time, followed by the vanilla. Mix until well combined, an additional 2 minutes. Stop the mixer and scrape down the sides of the bowl.
Add half of the flour mixture and stir to combine. Add the second half along with the crushed Oreos and stir to combine.
Add the dark and white chocolate and stir until just combined.
Using a large cookie scoop, measure out 2.5 ounce portions of dough and place on a rimmed baking sheet. Press more crushed Oreos on top if the dough.
Continue with the remaining dough. Cover the dough with plastic wrap and chill in the refrigerator for 2 to 24 hours.
When ready to bake, remove the cookies from the refrigerator and preheat the oven to 350°F. Place the cookies on a parchment paper lined baking sheet, at least 2 inches apart.
Bake up to two sheets at a time for 12 to 15 minutes. Rotate the baking sheets halfway through, giving the pan a quick bang on the oven rack to encourage spread. The cookies are done when the edges are set but the centers are still soft.
Give the baking sheet another rap on the counter or oven rack when done, then allow the cookies to cool on their baking sheets for 5 to 10 minutes before transferring them to a wire rack.