Brown the butter: In a small saucepan over medium heat, place half (1/2 cup) the butter. Swirl the pan as it cooks; the solids will turn golden brown (very quickly — watch it the whole time) and it’ll smell nice and nutty, about 3-5 minutes. Put the browned butter in a medium bowl or the bowl of a stand mixer, scraping in brown bits from the bottom of the pan. Add the remaining 1/2 cup butter to the bowl and let sit 10 minutes to let the browned butter cool. While the butter cools, measure out the rest of your ingredients.
Make the dough: Add 3/4 cup brown sugar and 1/2 cup granulated sugar into the butter and beat in using a stand mixer or hand mixer — low speed at first so butter doesn’t fly everywhere, then increase to medium high — until creamy and increased in volume by 50% or so, about 6 minutes. Add 1 egg, 1 egg yolk and 2 teaspoons vanilla and beat together for about a minute.
In a separate medium-sized bowl, mix together the dry ingredients: 2 teaspoons cornstarch, 1/2 teaspoon salt, 1/2 teaspoon baking powder, 1/4 teaspoon baking soda, 1 teaspoon cinnamon, 1 teaspoon ground cardamom, 1 1/2 cups rolled oats and 1 cup flour. Add to the bowl with the wet ingredients and mix at low speed until just combined. Stir in 1 cup cherries and 1 cup chopped pecans.
Divide dough into 12 balls, about 100 grams or 1/2 cup each. Chill the dough in the fridge for at least 1 to 2 hours, or freeze 30 minutes, until the dough is solid and chilled throughout.
Bake the cookies: Toward the end of the chilling time, preheat oven to 350°F. Place cookies on a parchment or silicone-lined baking sheet about 3 inches apart, no more than 4 cookies staggered on a baking sheet, as they spread quite a bit. Bake 17 to 19 minutes, until the edges are browned, there are some browned spots on the top of the cookies, and the center is slightly puffed but still quite soft.
Remove from the oven. Immediately out of the oven, use a round container with 4- to 5-inches diameter, like a deli container or a mason jar lid or a ring mold, and use the edge of the container to gently push the edges of the cookie toward its center. Once the cookie is “squished” up enough to fit inside the container, swirl the cookies around the inside of the container to get them perfectly round. Sprinkle with a little flaky sea salt and let rest on the baking sheet for 10 minutes, then place on a rack to cool completely. Repeat with remaining dough. (Note: Don’t place cookie dough on a hot baking sheet, use a different one from the one just out of the oven or cookies will bake unevenly). Keep leftovers at room temperature in an airtight container for a few days. The unbaked dough also freezes quite well, kept in an airtight container.