Preheat oven to 350 degrees F. Grease an 8-inch square baking pan and dust it with flour, knocking out any extra. Set aside.
Combine the egg, egg yolk, sour cream, lemon zest and vanilla in a large glass measuring cup and lightly beat. Combine 1¼ cups flour, baking powder, baking soda, and salt in a medium mixing bowl.
Combine the butter and sugar in a large mixing bowl and cream with an electric mixer on medium-high speed until fluffy, about 3 minutes, scraping down the sides of the bowl once or twice as necessary.
With the mixer on medium-low speed, pour the egg mixture into the bowl in a slow stream, stopping the mixer once or twice to scrape down the bowl.
Turn the mixer on low speed and add the flour mixture, ½ cup at a time, scraping down the sides of the bowl after each addition. After the last addition, mix for 30 seconds on medium speed.
Combine the strawberries and remaining ¼ cup of flour in a medium bowl and toss to coat. Fold the flour covered berries along with the chocolate chips into the batter with a rubber spatula.
Scrape the batter into the prepared pan and smooth the top with a spatula. Bake the cake until golden and a toothpick inserted in the center comes out clean, 45 to 50 minutes. Let the cake cool completely. Cut into 9 squares and serve.