Preheat the oven to 340°F (170°C). Line the base of an 8-inch round cake pan with parchment and lightly grease the sides.
Toast the pistachios: Spread the 2/3 cup pistachios on a small baking sheet and warm in the preheating oven for 6–8 minutes, just until fragrant. Cool completely. Finely chop 2 tablespoons and set aside for garnish. Pulse the remaining pistachios with 1 tablespoon of the sugar in a mini processor until the texture of fine meal (avoid overprocessing into paste).
Make lemon sugar: In a large mixing bowl, rub the lemon zest into the remaining sugar with your fingertips until the sugar feels damp and aromatic, 30–45 seconds.
Whisk the dry ingredients in a separate bowl: flour, baking powder, baking soda, salt, cardamom, and the pistachio meal you just made.
Build the wet base: To the lemon sugar, add the egg and yolk and whisk until thickened and paler, about 1 minute. Slowly stream in the oil while whisking, then mix in the honey, yogurt, milk, vanilla, and lemon juice until smooth.
Combine: Sprinkle the dry mixture over the wet and fold with a spatula just until no dry spots remain. Let the batter rest for 10 minutes to hydrate for a softer crumb.
Bake: Scrape the batter into the prepared pan and smooth the top. Bake 24–30 minutes, or until the center springs back and a tester comes out clean. Cool 10 minutes in the pan, then turn out to a rack and let cool completely.
Whip the frosting: In a cold bowl, whip the heavy cream to soft peaks. In another bowl, beat the softened cream cheese with powdered sugar, lemon juice, zest, vanilla, and a tiny pinch of salt until creamy and lump-free. Fold the whipped cream into the cream cheese mixture until smooth. Chill the frosting for 10 minutes to slightly firm up.
Finish: Spread the frosting over the cooled cake in gentle swoops. Sprinkle with the reserved chopped pistachios and, if you like, a thin drizzle of honey. Slice and serve.