Heat the oven to 350°F. Grease a 9x13-inch baking pan and line the bottom with parchment, leaving an overhang for easy removal.
Grate the lemon zest into the sugar and rub with your fingertips until fragrant; this infuses the sugar with lemon oils and brightens the cake.
In a large mixing bowl, beat the softened butter and the lemony sugar together until pale and fluffy (about 2–3 minutes). Add the oil in a slow stream while mixing until incorporated.
Add the eggs one at a time, beating after each until the mixture is smooth. Stir in the Greek yogurt and vanilla until combined, then fold in the poppy seeds if using.
Whisk the flour, baking powder, baking soda and salt together in a separate bowl. Gently fold the dry ingredients into the wet mixture with a spatula until just combined—don’t overmix or the cake will be dense.
Spread the batter evenly into the prepared pan and smooth the top. Bake for 30–38 minutes, or until a toothpick inserted into the center comes out with moist crumbs but no raw batter.
While the cake is baking, whisk the lemon curd and sweetened condensed milk together until smooth and pourable. Reserve 1–2 tablespoons for decoration.
Remove the hot cake from the oven and immediately poke holes all over the top using the handle of a wooden spoon or the end of a rolling pin. Pour the warm lemon mixture evenly over the cake, pressing slightly so the liquid settles into the holes. Let the cake cool on a rack for about 30 minutes, then refrigerate for at least 30 minutes to allow the drizzle to set.
For the topping: put the cold heavy cream and cold mascarpone into a chilled bowl with the sifted powdered sugar and vanilla. Whip with an electric mixer on medium-high until soft peaks form, about 2–3 minutes. Taste and adjust sweetness if desired.
Spread the mascarpone whipped cream over the chilled cake. Dot the reserved lemon drizzle on top and use a small offset spatula or skewer to swirl it into a gentle marbled pattern. Pipe a few rosettes if you like and finish with lemon slices and mint.
Keep the finished cake refrigerated until serving. For clean slices, chill briefly and cut with a hot, dry knife wiped between cuts.