To make the curd, place the granulated sugar and lemon zest in the bowl of a food processor fitted with the blade and pulse several times to combine. Add in the powdered sugar, eggs, egg yolk and lemon juice and run the machine for 30 to 60 seconds or until combined. Add in the flour and run the machine for 30 seconds more.
Strain the mixture into a heavy-bottomed saucepan over medium heat. Add the basil and butter and stir constantly for 6 to 8 minutes or until thickened. Strain once again if the mixture developed any lumps and immediately place in a small, heat-safe bowl and cover with plastic wrap, directly on the surface of the curd. Place in the fridge to chill completely.
To make the cookies, if your butter isn’t super soft, heat it in the microwave on high for 5 to 10 seconds. In an electric stand mixer fitted with the paddle, add the butter, powdered sugar and lemon zest and run the machine on medium for 4 minutes or until smooth. With the machine on low, add in the egg, egg yolk, milk, vanilla and salt and run the machine on low for several minutes more. Make sure to scrape down the sides and bottom of the bowl so that everything is incorporated. The mixture may look curdled. Dump in the flour and run the machine on the lowest setting until the dough comes together.
Place a silicone baking mat on the counter and then place a piece of parchment paper on top, lightly flouring it. Set the dough on top and sprinkle with a bit more flour. Rub some flour over your rolling pin and roll the dough to 1/4-inch (0.7-cm) thickness. Use the parchment paper to transfer it onto a baking sheet and chill in the freezer for 15 to 20 minutes or in the refrigerator for 30 to 40 minutes.
Preheat your oven to 350°F (177°C) and line several baking sheets with parchment paper. Grab a 1 3/4-inch (4.5-cm) round cookie cutter and a 1 1/4-inch (3.3-cm) round cookie cutter. Cut as many 1 3/4-inch (4.5-cm) circles out of the dough as you can. Re-roll the scraps and cut as many from that dough. Use the 1 1/4-inch (3.3-cm) cutter to cut smaller circles out of half of the 1 3/4-inch (4.5-cm) circles. Place all of the cookies, including the small circles from the cut-out cookies, on the prepared baking sheets with 1 1/2 inches (3.7 cm) between each cookie, and freeze for 10 minutes or refrigerate for 20 minutes.
Bake one sheet at a time in the center of the oven for 8 to 9 minutes. Let cool on the baking sheet. To assemble, for the larger cookies, smear some curd on the solid circle and top with an ‘O’ shaped cookie. For the smaller cookies, smear some curd on one cookie and top with a same-sized cookie.