In a large bowl, cream together the butter and both sugars with a hand-held mixer on medium speed (or mix by hand with a wooden spoon) until light and fluffy, about 5 minutes. Beat in the vanilla until thoroughly combined, scraping down the sides of the bowl.

In a medium bowl, sift together both flours, the baking powder, and the salt. With the mixer on low speed, slowly blend the dry ingredients into the wet ingredients, scraping down the sides, until the flour is completely blended and the dough is homogenous.

Scrape the dough from the bowl onto a sheet of plastic. Wrap well and press down to form a 1/2-inch-thick square. Refrigerate the dough until it's firm but still pliable, about 20 min. Heat the oven to 350 degrees F. Prepare two sheets of parchment or waxed paper, each at least 11x11 inches. If using waxed paper, grease an 8x8-inch metal or glass baking pan with butter.

When the dough is firm, unwrap it and put it between the sheets of parchment or paper. Roll the dough to an approximate square, slightly larger than 8x8 inches and about 1/4 inch thick. Remove the top sheet of parchment or paper, trim the dough with a dull knife to an 8x8-inch square, and, if using parchment, put it and the dough into an 8x8-inch baking pan. If using waxed paper, flip the dough into the greased pan and peel off the paper. Press the dough into the bottom of the pan, letting the excess parchment come up the sides (trim it to about 1 inch above the rim). The dough should be an even thickness all around but it needn't be perfectly smooth. Bake until the shortbread is light golden on top, 25 to 30 min.; in a glass pan, look for a golden brown color on the bottom. Remove the pan from the oven, but keep the heat set to 350 degrees F as you make the lemon curd.

In a medium saucepan, heat the lemon juice, butter, and cream to just under a boil; the butter should be melted. Remove from the heat.

In a medium bowl, whisk together by hand the sugar, eggs, and yolks until combined. Whisk in a bit of the hot liquid and then gradually whisk in a bit more until it's all added. This technique, called tempering, heats the eggs slowly and gently so they don't curdle.

Pour the mixture back into the saucepan and heat on medium, stirring constantly with a wooden spoon, scraping the bottom and sides of the pan to keep the eggs from scrambling. Cook until the lemon curd coats the spoon thickly enough to leave a line when you draw your finger through, 5 to 8 minutes. Remove from the heat and strain through a fine sieve. Stir in the salt and vanilla.

Pour the curd over the baked shortbread and smooth it evenly with a spatula, if needed. Bake until the curd has set and jiggles like firm jello, 15 to 20 minutes. Let cool to room temperature. Gently tug on the parchment on all sides to loosen the bars from the pan. Lift them out and onto a cutting board and refrigerate until the curd has completely set, at least 4 hours. Trim the sides for a cleaner look and cut into 16 pieces.

Make Ahead: Shortbread dough can be wrapped in plastic and frozen for up to two months. When you want to make lemon bars (or any other shortbread-based recipe), just let the dough thaw overnight in the refrigerator.