In a large mixing bowl, cream the butter and sugar on medium speed until light and fluffy, about 3 minutes. Scrape down the bowl, then add the egg and beat until fully incorporated, about 30 seconds. Add the cream cheese and beat for 1 minute, scrape down the bowl once more, then beat in the vanilla extract, almond extract, and lemon zest until incorporated, about 30 seconds.
In a separate bowl, whisk together the flour, baking powder, and salt. With the mixer running on low speed, slowly add the dry ingredients to the butter-sugar mixture, mixing just until fully incorporated and a soft dough is formed.
Divide the dough into quarters. If using the dough within 24 hours, roll each quarter to a 1/4-inch thickness in between two sheets of parchment paper (alternatively, you can store the dough in a ball or disk and roll out when ready to cut and bake). Be careful not to roll the dough any thinner or the cookies will not be as soft. Keeping the dough flat, transfer it to the refrigerator and chill for at least 1 hour or overnight. Repeat with the remaining dough (it is fine to stack the dough "flats" on top of each other to save space). If you will not be using the dough within 24 hours, divide the dough in half, pat each into a disk, then wrap each disk tightly in plastic and freeze in a ziptop bag for up to 1 month. Let thaw in the refrigerator overnight, then proceed with the recipe as directed.
When ready to bake, preheat the oven to 350 degrees F and line baking sheets with parchment paper or a silicone baking mat. Remove one of dough flats from the refrigerator. (If you stored the dough in a ball or disk, roll it to a 1/4-inch thickness now, between two layers of parchment paper.) Remove the top layer of parchment paper, then with cookie cutters, slice the cookies into desired shapes. Transfer to the prepared baking sheet. Repeat with remaining dough.
Bake the cookies for 9-11 minutes, until barely lightly golden brown and the edges are dry. The cookies will look slightly underbaked. Let cool completely.
To make the frosting: In a small bowl, stir together the powdered sugar and 3 tablespoons milk as best as you can (the frosting will be very solid at this point). Beat in the corn syrup and almond extract until the frosting is smooth and glossy. If the frosting seems too thick, add more milk, one teaspoon at a time, until you reach your desired consistency. If making colored frosting, add a few drops of food coloring until you reach your desired color intensity. Frost and decorate the cookies as desired. Let the frosting set for a few minutes, then enjoy!
Store leftover baked cream cheese sugar cookies in an airtight container at room temperature, separating the cookie layers with wax or parchment paper, for up to 4 days or freeze frosted or unfrosted for up to 2 months.
Dough can be prepared through Step 3 and stored covered in the refrigerator for up to 1 day or frozen for up to 3 months. Let stand at room temperature until soft enough to scoop, then bake as directed (if frozen, thaw overnight in the refrigerator first).
Shaped, unbaked cookies can be frozen for up to 3 months. To freeze: place the unbaked cookies on a parchment-lined baking sheet, freeze until hardened, then transfer to a ziptop bag. Bake directly from frozen, adding a minute or two to the baking time.
If you want to make a monster batch of cookies, this recipe can be doubled.
Do not substitute the corn syrup—it is key for icing that sets up but still stays soft. Normally I'm all for baking without it, but here it is worth it.