Make the cheesecake center: in a medium bowl, beat the softened cream cheese and powdered sugar with a hand mixer (or whisk vigorously) until silky and lump-free. Stir in the lemon zest. Transfer the mixture to a small bowl, cover, and chill in the refrigerator while you prepare the dough so it firms up — about 15 minutes.
Heat the oven to 350°F (175°C). Line two baking sheets with parchment paper and set aside.
In a large bowl, cream the room-temperature butter with the brown sugar and granulated sugar until smooth and slightly fluffy (2–3 minutes) using a hand mixer or stand mixer fitted with the paddle. Scrape down the bowl.
Add the whole egg, egg yolk and vanilla to the butter mixture and beat until combined. Stir in the baking soda and salt until evenly distributed.
Lower the mixer speed or fold in the flour by hand with a sturdy spatula until just incorporated — do not overmix. Fold in the chocolate chips so they are evenly dispersed through the dough.
Portion the cookie dough into 24 equal pieces (about 1 1/4 tablespoons per piece). Roll each piece into a ball and place on the prepared sheet. Flatten each ball into a 2-inch disk in your palm.
Scoop about 1 teaspoon of the chilled cheesecake filling onto the center of each flattened dough disk. Wrap the dough up around the filling and pinch to seal, rolling gently to form a smooth ball so the filling is completely enclosed.
Arrange the filled cookie balls 2 inches apart on the baking sheets. For a slightly neater shape, chill the assembled cookies on the tray in the refrigerator for 8–10 minutes before baking.
Bake one sheet at a time in the preheated oven for 11–12 minutes, or until cookie edges are golden and the tops look set but still soft. Rotate the pan halfway through for even browning.
Remove from oven and let cookies rest on the baking sheet for 3–4 minutes to finish setting, then transfer to a cooling rack to cool completely so the cheesecake center firms up.