In a large mixing bowl, use your fingertips to massage lemon zest into sugar until fully incorporated. Add brown sugar and butter. Use a handheld mixer on medium-high speed to combine until light and fluffy (2-3 minutes). Switch to low speed and mix in the non-dairy milk and vanilla extract.

Add flour, oats, baking soda, and salt. Combine on low speed until a soft cookie dough forms. Refrigerate dough for 30 minutes. Use a cookie scoop to portion dough onto a parchment-lined baking sheet. Place in freezer for 15 minutes to further firm up (setting the dough in the fridge and freezer is is crucial to maintain cookie shape and reduce spread). 

Preheat oven to 350°F. Use a 1.5-2 tbsp cookie scoop to portion out dough on to a lined baking sheet, about 6 to a sheet. These cookies will spread quite a bit, but that's what creates a soft and chewy texture. Bake for 10 minutes until golden brown and crisp on the edges. They may appear slightly underdone in the middle – that's ok! It will set once cooled. 

Cool cookies for 10 minutes on the baking sheet before transferring to a wire cooling rack. Allow cookies to completely cool. 

In a large mixing bowl, use a handheld mixer on medium-high speed to cream together the cream cheese, butter, vanilla and lemon extract, lemon zest, and salt until light and fluffy, about 2 minutes. Add the powdered sugar 1 cup at a time, mixing each until combined. It should have the consistency of marshmallow fluff, spread easily, and not melt off the sides of the cookie. Add ¼ cup at a time of powdered sugar, if the texture is too runny.

Flip over half the cooled cookies and use an offset spatula or spoon to spread the filling on top of the flipped cookies. Press the remaining cookies on top to make sandwiches. Refrigerate the cookies in an airtight container for up to 5 days.