Place the quartered figs in a medium, heavy-bottomed saucepan. Add the packed brown sugar, grated lemon peel, kosher salt, and vanilla extract, then stir to combine.
Set the saucepan over medium heat. Cook until the figs begin releasing their liquid and the mixture reaches a gentle boil.
Lower the heat and allow the fruit mixture to simmer for 45 minutes. Stir periodically, using the back of a wooden spoon to crush any particularly large pieces of fig.
When the filling has thickened and the figs are tender, remove the saucepan from the heat and leave the mixture to cool.
For a more uniform filling, process the cooled mixture with an immersion blender, countertop blender, or food processor. Alternatively, retain a few pieces of fruit for a chunkier, rustic consistency.
Preheat the oven to 350°F (175°C).
Line a 9 × 13-inch baking pan with aluminum foil, leaving enough excess foil over the sides to serve as handles when removing the baked bars. Lightly coat the foil with cooking spray.
Place the unsalted butter in a large microwave-safe mixing bowl and melt it completely.
Add the rolled oats, all-purpose flour, packed brown sugar, baking powder, salt, and vanilla extract to the melted butter.
Mix until the ingredients are thoroughly incorporated and no visible patches of dry flour remain. The finished mixture should have a dense, crumbly consistency. Use clean hands to complete the mixing process when necessary.
For a deeper oat flavor, the rolled oats may be lightly toasted in a dry skillet before they are incorporated into the crust mixture.
Transfer the remaining oat mixture to the prepared baking pan. Press it firmly and evenly across the bottom, using your hands or the flat base of a drinking glass to create a compact crust.
Spoon the cooled fig filling over the crust and spread it into an even layer that reaches the edges.
Scatter the reserved oat mixture across the fig filling. Distribute the crumbs evenly while allowing small portions of the fruit layer to remain visible.
Bake the assembled bars at 350°F (175°C) for 30–35 minutes. The bars are ready when the fig filling is bubbling and the oat topping has developed a golden-brown color.
Allow the bars to cool completely in the baking pan.
After cooling, grasp the overhanging foil and carefully lift the entire slab from the pan.
Cut into individual bars and serve.