Preheat oven to 375 degrees. Position a rack in the middle of the oven. Using nonstick spray, grease 2 baking sheets, or line them with parchment paper.
In a medium bowl, sift together the flour, baking soda, baking powder, and salt.
Using a handheld electric mixer, cream the butter and sugars together in a large bowl until light and fluffy, about 3 minutes. Add the eggs, 1 at a time, beating until incorporated. Beat in the vanilla. Mix in the dry ingredients with the mixer on low speed until just combined. Stir in the chocolate and cherries with a wooden spoon.
Scoop by the heaping tablespoonful onto the baking sheets. Bake until golden and chewy, about 12-15 minutes. Transfer the cookies to a rack to cool. Store in an airtight container.