Pulse the oats in a food processor until some of it is fine and some of it is still very coarse. Dump the oats into a medium bowl and whisk in the flours, baking soda, baking powder and salt. In a large bowl, beat the butter and the sugars with a mixer at medium speed until very light, about 5 minutes. Beat in the eggs, one at a time, until well blended. Beat in the vanilla.
At low speed, add the flour mixture and mix until just combined. Using a rubber spatula or a wooden spoon, fold in the cherries, nuts and white chocolate chips. Cover the bowl with plastic wrap and chill for at least 24 hours and up to 48 hours.
Position the racks in the upper and lower thirds of the oven and preheat the oven to 350 degrees F. Line baking sheets with parchment paper or silicone mats.
Roll about 2 teaspoons of dough into balls and place them on the lined baking sheets, leaving about 1 inch of space between them.
Bake until the cookies are golden brown around the edges and the centers are pale golden and still soft, about 9 minutes. Let cool on wire racks for about 10 minutes, then transfer the cookies with a spatula to the racks to cool completely (or for as long as you can wait!).
To make sure the cookies bake evenly, switch the position of the baking sheets (meaning move the cookies on the upper rack to the lower rack and vice versa) and rotate the sheets from front to back halfway through the baking time.