Preheat the oven to 325° F. Line a baking sheet with parchment paper and set aside.
Cream butter and brown sugar on low speed in mixer bowl (stand or handheld) until combined. Increase speed to medium to medium-high and beat for about 4-5 minutes, until very fluffy and light in color. Scrape down sides and bottom of bowl.
On low speed, add eggs, vanilla, and rum (if using) and mix to combine. Scrape down sides and bottom of bowl and blend for about 1 minute on medium speed.
Whisk together the flour with the baking soda and salt in a medium bowl. Gradually add it to the butter/egg mixture on low, beating only until just combined.
Stir in the white chocolate chips, strawberries, and blueberries by hand. The dough will be very thick and you may need to use your hands to combine everything fully.
Scoop with 3 tablespoon cookie scoop and roll into balls. Roll into turbinado sugar, if desired, and place 2 inches apart on prepared baking sheet. Flatten to about ½-inch thick with the palm of your hand.
Bake for 12 minutes. Cool on wire rack. Store in an airtight container at room temperature for up to 5 days.
Notes
Cookies will keep in an airtight container for up to 5 days, or can be frozen for up to 3 months. Unbaked cookie dough may also be frozen. Shape into balls and dredge in turbinado sugar. Place on baking sheet and freeze until solid. Store in large ziptop bag until ready to bake. Can be baked from frozen, adding 5-7 minutes to bake time.