Preheat the oven to 350 F. Prepare two baking sheets by either spraying with baking spray, lining with parchment paper, or a silicone baking mat.
With an electric mixer using a paddle attachment, combine the butter, and white sugar, then beat on high speed until mixture is light and fluffy, about 3 minutes, scraping the sides of the bowl occasionally.
Add egg, vanilla, and salt and mix well on medium speed. Then add lemon zest, juice, and poppy seeds and mix until combined.
Add in instant pudding mix, baking soda, and mix. Then add in flour one cup at a time until well mixed.
Scoop cookie dough in ⅓ cup portions onto a cookie sheet, 6 to a pan to allow for spreading. Flatten the tops of each dough ball slightly to make a thick round hockey puck shape.
Bake for 11-12 minutes, until just barely golden brown and beginning to crack. Cool completely before frosting.
For the Frosting:
Using an electric mixer, mix together softened cream cheese and butter on medium speed for 1-2 minutes until creamy and fluffy.
Add in mashed blackberries, vanilla, and cornstarch. Mix in powdered sugar one cup at a time on low speed until frosting reaches desired consistency. Refrigerate for 15-20 minutes to firm up if needed before piping.
Transfer to a piping bag with a round frosting tip and pipe onto cooled cookies in a spiral pattern, starting in the middle and working outward.
Spoon 1 Tablespoon of blackberry jam onto the middle of each cookie and spread flat, careful not to mix into the frosting.
Notes
Once frosted, these cookies can be refrigerated in an airtight container for 3-4 days. Cream cheese frosting should be refrigerated if not served/consumed within 3-4 hours after making and assembling cookies.