Preheat oven to 350°F.
Combine the lemon zest and sugar. In a large bowl, combine the lemon zest and sugar by pinching them together with your hands to release the lemon flavor into the zest.
Whisk in the eggs. Crack the eggs into the bowl with the sugar and whisk vigorously until well combined and foamy.
Whisk in the remaining wet ingredients. Pour the olive oil, honey, lemon juice, and vanilla into the same bowl. Whisk to combine.
Add the dry ingredients. Sprinkle the flour, baking powder, and kosher salt in an even layer on top of the wet ingredients. Whisk until just combined – some lumps remaining are okay.
Bake. Pour the cake batter into a greased and lined 8x8 round cake pan. Tap the pan gently on the table to help release any air bubbles. Carefully transfer to a preheated 350°F oven for 25 minutes and bake until golden brown on top and a tester comes out clean.
Make the lemon syrup. While the cake bakes, in a small saucepan, combine the honey, water, and lemon juice. Warm over medium heat and cook for just a few minutes until all combined.
Top with lemon syrup. When the cake comes out of the oven (still in the pan), let it sit for about 5 minutes. Then carefully and slowly spoon the lemon syrup over top (or brush using a pastry brush). Let it cool the rest of the way in the pan.