Heat the oven to 350°F and line a baking sheet with parchment paper. If you have a cooling rack ready, set it nearby.
In a large bowl, beat the softened butter with the granulated and powdered sugars until the mixture is light and fluffy, about 2–3 minutes using an electric mixer on medium speed.
Add the egg, lemon juice, and lemon zest to the butter-sugar mixture and mix until fully incorporated and smooth.
Whisk together the flour, baking powder and salt in a separate bowl. Fold the dry ingredients into the wet mixture with a spatula just until no streaks of flour remain. If using, stir in the poppy seeds. Do not overmix.
Using a 1/4-cup scoop (about 4 tablespoons), portion 6 mounds of dough onto the prepared sheet, spacing them well. Press each mound gently to flatten to about 3/4-inch thickness — these are large bakery-style cookies.
Place the baking sheet in the refrigerator for 15–20 minutes to firm the dough slightly; this helps the cookies hold their shape during baking.
Bake for 12 minutes, or until the edges are set and the tops are lightly golden. Rotate the pan once halfway through for even browning. Remove from oven and let the cookies cool on the baking sheet for 8 minutes before transferring to a rack to cool completely.
While the cookies cool, prepare the blackberry frosting. Mash the blackberries and press through a fine-mesh sieve to remove seeds; you should have about 1 tablespoon of strained juice. If it’s very loose, simmer the juice in a small saucepan over low heat for 1–2 minutes to reduce slightly, then cool.
Beat the softened cream cheese and butter together until smooth. Gradually add the powdered sugar and beat until creamy. Mix in the strained blackberry puree, lemon zest and vanilla, beating just until combined and fluffy.
Spread or pipe a generous layer of the blackberry cream cheese frosting onto each cooled cookie. Top each with a small spoonful of blackberry jam and, if desired, a tiny pinch of flaky sea salt to accent the flavors.
Serve immediately or chill for 10–15 minutes to set the frosting slightly. Store leftovers in an airtight container in the refrigerator for up to 4 days.