Preheat oven to 350°F. Line 2 sheet pans with parchment paper.
In the bowl of a stand mixer fitted with the paddle attachment, add the sourdough starter, butter, oil, granulated sugar, and powdered sugar. Sprinkle the salt, baking powder, and baking soda over the ingredients. Mix until creamy, 1-2 minutes. Scrape down sides of bowl as needed.
Add the sour cream, eggs, vanilla, and almond extract. Mix until combined, about 1-2 minutes.
Add the flour and mix until combined. Scrape down sides of bowl as needed. Do not overmix.
Scoop the cookie dough using a cookie scoop or spoon. Use about 3 Tablespoons dough for each cookie. Roll dough into balls. Place 1/2 cup granulated sugar in a small bowl. Roll dough in the sugar, coating it completely. Place cookies on the baking sheet 2-3 inches apart. You can fit 8 cookies per baking sheet. Using the flat bottom of a glass, press the center of each cookie, flattening it. Don't flatten too thin - you want them about 1/4 to 1/2-inch in thickness. Dip the bottom of the glass in the sugar between pressing each cookie. This will keep it from sticking to the dough.
Bake cookies for 8-10 minutes, until edges are set. The center will look soft, but that's ok. Let the cookies cool on the baking sheet for 10 minutes. Then transfer to a wire cooling rack to cool completely.