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PdfHeat oven to 350 degrees. Line an 8-inch square baking pan with parchment paper, leaving a 2-inch overhang on two opposite sides.
In a large bowl, whisk together the yogurt, brown sugar, applesauce, eggs, oil and vanilla. Add the oats, flour, cinnamon, baking powder, baking soda and salt. Fold by dragging a large flexible spatula under and over the mixture, scraping the sides and bottom of the bowl as you go, until no dry streaks are visible. Transfer batter to the prepared pan and bake until a toothpick inserted into the center comes out with moist crumbs attached, 27 to 30 minutes.
Transfer to a rack to fully cool. Once cool, using a paring knife, separate the two edges without parchment from the pan. Then using the parchment overhang, lift out of the pan and transfer to a cutting board. Cut into 16 pieces with a serrated knife.