Drain the maraschino cherries through a fine mesh strainer, reserving 2 tablespoons of juice in a small bowl. Spread cherries on paper towels, press dry, and finely chop into small pieces; set aside.
Preheat the oven to 350°F (175°C) and position racks in the upper-middle and lower-middle positions. Line two large baking sheets with parchment paper or silicone baking mats.
In a medium bowl, whisk together the flour, baking soda, cornstarch, and salt until evenly combined. Set aside.
In a large bowl, beat the softened butter, granulated sugar, and brown sugar together on medium-high speed for 3 to 4 minutes until light, pale, and fluffy. Scrape down the sides of the bowl as needed.
Add the egg, egg yolk, almond extract, vanilla extract, and reserved maraschino cherry juice to the butter mixture. Beat on medium speed for 1 to 2 minutes until fully combined and creamy.
Add the flour mixture to the wet ingredients and mix on low speed until just barely combined, stopping as soon as a few flour streaks remain. Do not overmix.
Using a rubber spatula, gently fold in the chopped maraschino cherries and cherry baking chips until evenly distributed throughout the dough.
Use a medium cookie scoop (about 1.5 tablespoons) to portion dough balls onto the prepared baking sheets, spacing them 2 inches apart. Optionally press a few extra cherry chips on top of each ball.
Bake for 10 to 12 minutes, rotating pans top to bottom at the 6-minute mark, until edges are set and lightly golden but centers still look slightly underdone. Do not overbake.
Cool cookies on the pans for 5 minutes, then transfer to a wire rack to cool for at least 10 more minutes before serving. Cookies firm up and develop their chewy texture as they cool.