Bring the figs and 2 cups of the apple juice to a simmer in a medium saucepan over medium heat. Cook, stirring occasionally, adding more juice as necessary, cup at a time, until the figs are puffed and soft and most of the liquid has been absorbed and the remaining liquid is thick and syrupy, 30 to 40 minutes. Transfer the figs and the remaining apple juice to a food processor and puree until very smooth, about 1 minute, scraping down the sides of the bowl with a rubber spatula as necessary. Transfer to a medium bowl and set aside at room temperature until completely cool, about 45 minutes.
Adjust an oven rack to the lower-middle position and heat the oven to 350 degrees. In a 9-inch square baking pan, fit one sheet of foil or parchment (large enough to overhang the sides of the pan) into the bottom of the pan, pushing it into the corners and up the sides of the pan (the overhang will help in the removal of the baked bars). Fit the second sheet in the pan in the same manner, perpendicular to the first sheet. Spray the foil with vegetable oil spray.
Whisk the flour, oats, cinnamon, baking soda, and salt together in a medium bowl and set aside. In the bowl of a standing mixer fitted with the paddle attachment, beat the butter, brown sugar, and vanilla on medium speed until light and fluffy, about 3 minutes. Turn the mixer to low speed, add the dry ingredients, and beat until the mixture is well combined and resembles moist sand, about 2 minutes. Transfer half of the mixture to the prepared pan and use your hands to press the crumbs evenly into the bottom. Using a rubber spatula, spread the cooled fig puree evenly over the bottom crust. Sprinkle the remaining crumbs evenly over the puree and press lightly to adhere.
Bake until the fig puree bubbles around the edges and the top is golden brown, 40 to 50 minutes, rotating the pan from front to back halfway through the baking time. Cool on a wire rack to room temperature, about 2 hours. Remove the bars from the pan using the foil handles and transfer to a cutting board. Cut into 1-inch squares and serve.