Heat oven to 350ºF with rack set in middle position. Grease 9-inch springform pan (if your springform pan is prone to leaking, wrap the base in a piece of foil).

Whisk together flour, baking powder, and salt in bowl; set aside.

Using stand mixer fitted with the whisk attachment, whip eggs on medium speed until foamy, about 1 minute. With mixer running, add sugar, lemon zest, and vanilla. Increase speed to high, and whip until mixture is fluffy and pale yellow, about 3 minutes.

Reduce speed to medium and, with mixer running, slowly stream in oil; mix 1 minute. Reduce speed to low and add half of the flour; mix until incorporated, about 1 minute, scraping down side of bowl as needed. Add milk and mix until combined, about 30 seconds. Add remaining flour mixture and mix just until incorporated, about 1 minute, scraping down side of bowl as needed.

Transfer batter to prepared pan; evenly sprinkle top with about 2 tablespoons sugar. Bake cake until top is deep golden brown and a toothpick inserted in the center comes out with few crumbs attached, 48–55 minutes.

Transfer pan to wire rack and let cool for 20 minutes. Loosen and remove side of pan; let cake cool completely, about 1½ hours. Loosen cake from the pan base, transfer to a serving plate, and cut into wedges; serve. 

Optional, top servings with whipped cream, candied lemon wedges, and a drizzle of good-quality olive oil.