Wash, peel, and cut the potato into 1" pieces.
Place in a saucepan in lightly salted water, and bring to a boil. Reduce the heat to a simmer and cook for about 15 minutes or until potato pieces are easily pierced with a knife. Drain, cover, and return to low heat for 5 minutes to dry a bit.
Preheat oven to 400F and set a rack in the top and bottom thirds. Line your cookie sheets/baking pans with Silpat or parchment and set aside.
Mash well with a masher and measure out 2/3 cup (6 oz) of mashed potato. Put in a bowl and save the rest of the potato for another purpose. I just make buttery mashed potatoes and eat them as a snack.
Stir the milk (cold is fine) into the reserved mashed potatoes and set aside.
Whisk the flour and baking powder together. Set aside.
Cream together the butter, sugar, salt, lemon zest and vanilla extract until light and fluffy, scraping the bowl as necessary.
Add the egg and beat until combined, scraping the bowl as necessary.
Mix in the mashed potato/milk mixture until combined, scraping the bowl as necessary.
Add the flour/baking powder mixture all at once and mix on low until combined. Scrape the bowl. The texture of your batter should be like very thick cake batter. It should flow a little bit but not be runny.
Portion cookies using a 2 oz scoop for large cookies or a 1 oz scoop for smaller cookies. Leave a good 2 1/2" between the cookies, especially if using 6 oz of milk. You may need to bake a 3rd round, so don't try to crowd the pans or the cookies will run into each other and end up with some flatter sides, and we want nice, round cookies.
Bake large cookies for about 16 minutes and smaller cookies for about 12 minutes. Rotate the pans and swap racks halfway through baking. The cookies are done when they are firm and barely starting to color on the tops and are a warm, deep golden brown on the bottoms.
Let cool on the sheets for a couple of minutes, and then carefully transfer them to racks to cool completely.