Sift flour once, measure, then add baking powder and salt. Sift all together two more times.
Cream butter; add sugar gradually, beating until light and fluffy. Add egg yolks, beating well. Add flour mixture alternately with the milk, beating after each addition until smooth. Stir in vanilla.
Pour batter into a greased and floured 8-inch square pan. Bake in a preheated 350° oven for 45 minutes, or until a wooden pick or cake tester inserted in center comes out clean. Cool for 10 minutes; carefully remove to a rack to cool thoroughly.
Combine egg white, sugar, water, and lemon juice in top of double boiler, beating with mixer until well mixed. Place over rapidly boiling water, beating constantly, and cook for 4 minutes, or until frosting will stand in peaks. Remove from the boiling water and add lemon peel, almond extract, and figs. Beat until thick enough to spread. Frosts top and sides of an 8-inch square cake.