Preheat your oven to 375 degrees F.
Prep a 9" springform pan by buttering the bottom and inner sides, then lining with parchment paper. You should use just enough butter to let the parchment paper stick onto the bottom and sides. When done, set aside.
Measure out your sugar into a stand mixer bowl or another large bowl.
Wash and dry two lemons. Then, zest them into the sugar. Once you've fully zested the lemons, rub the zest into the sugar using your hands. The sugar should become fragrant and lemon-y. Do this for about 20-30 seconds.
Add your butter to the sugar, and cream them together on medium speed using a stand mixer or a hand-held mixer. You want to see the butter and sugar completely incorporated, and the butter starting to get fluffy. This will depend on the temperature of your butter, but it'll usually take about 3-5 minutes.
Once the butter and sugar are creamed, add in the eggs, lemon juice, milk, and vanilla extract, and mix again on medium-low speed until everything is fully incorporated.
Then, add the flour, baking soda, baking powder, and salt, and mix by hand with a spatula until just combined.
Pour the cake batter into your prepared pan, and use the spatula to level out the top.
Bake the cake in the oven for about 30-40 minutes, or until a toothpick comes out clean. Be sure to check often - all ovens are different and cook times can vary. When done, set aside to cool in the springform pan.