To the bowl of a stand mixer fitted with the paddle attachment, beat the butter on low speed until smooth, 1 to 2 minutes. Add the sugars and beat on medium-high speed until creamed and well combined, 2 to 3 minutes. Stop, scrape down the sides of the bowl, and add the egg, cream, vanilla, and beat on medium-high speed until light and fluffy, about 3 minutes. Stop, scrape down the sides of the bowl, and add the flours, baking soda, salt, and mix until just combined, about 1 minute. Add the cherries, white chocolate chips, and beat momentarily with the mixer until combined or fold in by hand; don't overmix or the cherries will break.
Notes regarding the flours - Solely using all-purpose flour will work, but the cookies will not be as chewy or rise as well because bread flour creates chewier results and gives greater rise. I live in a dry climate and only need 1 3/4 cups flour total but if you are in more humidity or your dough is very moist or loose, or are exclusively using all-purpose flour, adding up to 1/4 cup of additional flour, for 2 cups total, is possible. The more flour, the more the cookies will stay domed and puffed while baking and the less likely they will be to spread. If you're uncertain, err on the side of more, and if the cookies are too puffy after baking, you can tap them with the back of a spoon a couple times as they cool on the baking sheets to flatten them. Regardless of using bread or all-purpose, this is a fairly sticky and tacky dough because of the cherries.
Using a 2-ounce or medium cookie scoop, form heaping mounds weighing 2 1/4-ounces each, a scant 1/4-cup measure; or divide dough into approximately 14 equal-sized pieces. Place dough mounds on a large plate, cover with plastic wrap, and refrigerate for at least 2 hours; up to 5 days. Do not bake these cookies with dough that has not been properly chilled because they will spread.
Preheat oven to 350°F, line a baking sheet with a Silpat Non-Stick Baking Mat, parchment, or spray with cooking spray. Place dough on baking sheet, spaced at least 2 inches apart; I bake a maximum of 8 per sheet. Bake for 8 to 10 minutes, or until tops have just set, even if slightly undercooked in the center, as cookies will firm up as they cool For crunchier cookies, extend baking time by 1 to 2 minutes. Allow cookies to cool on the baking sheet for 10 minutes before removing. Store cookies in an airtight container at room temperature for up to 5 days (I had success for 7 days), or in the freezer for up to 3 months. Alternatively, unbaked dough can be stored in an airtight container in the refrigerator for up to 5 days, or in the freezer for up to 3 months, so consider baking only as many cookies as desired and save the remaining dough to be baked in the future when desired.