Preheat the oven to 325°F / 165°C. Generously grease a 9 by 4-inch / 23 by 10cm Pullman pan (or a 10 by 5-inch / 25 by 13cm loaf pan), then line it with greased parchment paper. Set aside.
Put the lemon slices and vanilla seeds in the bottom of a 9-inch / 23cm baking dish, pour the simple syrup over the top, and cover tightly with aluminum foil. Bake until the lemons are very soft, 40 to 45 minutes. Set aside.
In a stand mixer fitted with the paddle attachment, cream the butter on medium-high speed until creamy and smooth, about 1 minute. Turn the mixer speed to low; add the sugar, vanilla extract, lemon extract, and lemon zest to the butter; and mix until incorporated. Turn the speed to medium-high and beat until light and fluffy, about 5 minutes. Scrape the bowl often for even incorporation.
Turn the speed to medium-low and add the eggs and egg yolks, one at a time, beating just until combined. Scrape the bowl after each addition.
In a separate bowl, whisk together the flour, baking powder, baking soda, and salt. Sift to remove any lumps.
Add one-third of the flour mixture to the butter mixture and mix on low speed, just until combined. Add half of the sour cream, mixing until incorporated. Repeat with another one-third flour, then the remaining sour cream, and finish with the final one-third flour, scraping the bowl and paddle after each addition.
Fold the lemon curd into the batter. It’s okay if there are a few streaks of curd left throughout the batter. Scrape the batter into the prepared pan. Smooth the top and tap the pan on the counter several times to release excess air bubbles. Lay five baked lemon slices over the top of the batter. Set the pan on a baking sheet.
Bake just until the top sets, about 45 minutes; the sliced lemons will have slipped beneath the surface. Lay another four lemon slices over the top and continue to bake until a tester comes out clean, 25 to 30 minutes more. As soon as you take the cake out of the oven, drizzle with 2 Tbsp of the lemon syrup from the baked lemons.
Let the cake cool in the pan for 20 minutes, then remove from the pan and set on a wire rack to cool completely before serving.
For the Lemon Curd: In a medium stainless-steel bowl, whisk together the egg yolks, sugar, lemon juice, lemon zest, butter, and salt. Put 1 inch / 2.5cm of water in the bottom of a double boiler or a medium saucepan and bring to a gentle simmer over medium heat. Place the bowl with the lemon mixture over the simmering water. Using a rubber spatula, stir the mixture constantly, making sure to clean the sides of the bowl as you go, until the lemon curd begins to thicken, about 10 minutes; it will be the consistency of smooth pudding. If there are any lumps in the curd, strain it through a fine-mesh sieve into a shallow container and cover with plastic wrap, pressed directly onto the surface, to prevent a skin from forming. In a large bowl, add enough ice cubes and cold water to create a bath for the container to sit in without the water breaching the sides. Set the container in the ice bath or place in the freezer until chilled, about 15 minutes, then transfer to the refrigerator for up to 5 days.