Preheat the oven to 350 degrees F (175 degrees C). Spray two 8-inch round cake pans with cooking spray.
To make the fig cake: Sift flour, salt, and baking powder together in a medium bowl; set aside.
Beat sugar and butter together in a large mixing bowl with an electric beater until fluffy. Add egg and beat well. Add flour mixture alternately with evaporated milk. Fold in vanilla, almond extract, and 1 cup chopped figs.
Spoon cake batter evenly into the prepared cake pans. Bake in the preheated oven until a toothpick inserted into the center comes out clean, 25 to 30 minutes. Cool cake layers on a wire rack.
To make the fig filling: Combine 2 cups chopped figs, brown sugar, water, and lemon juice in a saucepan. Bring to a boil. Reduce heat to a simmer and cook until thickened, about 20 minutes. Spread thinly between cooled cake layers and on top.
