Preheat the oven to 350°F. Line an 8x8 baking dish with aluminum foil.
Make the shortbread first: beat unsalted butter for one minute, then scrape and add powdered sugar and beat for two more minutes on high, scraping down every minute or so.
Add flour and salt, beating on low until just combined. Mixture will be very crumbly.
Press the shortbread dough into the bottom of the baking dish. Use the bottom of a glass to get it as even as possible. Prick with a fork, then bake for 18-22 minutes until the edges are just beginning to color.
Make the curd while the shortbread bakes: first, set a strainer over a mixing bowl next to the stove. In a small saucepan, whisk together eggs, egg yolks, and sugar. Once combined, whisk in lemon juice and salt.
Put the lemon mixture over medium-low heat and stir continuously for about 8 minutes or until the mixture has thickened enough to coat the back of a spoon. If it clumps while you're stirring, take off the heat.
Pour the curd through the strainer and stir in the chunks of butter. Keep stirring until everything has fully combined.
When the shortbread is done, take out of the oven and pour the curd on top. Return to the oven and bake for another 12-15 minutes, until the edges look set but the middle is still a bit wobbly.
Cool completely on the counter, and refrigerate overnight once it's room temperature.
Cut into bars only when it's very cold, just before serving. Wipe the knife clean in between slices, it helps with cleaner cuts.