Make the candied lemon slices by combining water and sugar in a medium saucepan. Bring to a boil and cook just until the sugar is dissolved. Add the lemon slices. Simmer until the lemon slices are translucent, roughly 30 to 40 minutes. Gently place the lemon slices on a silicone mat. Allow the lemons to cool and dry completely, overnight preferred.
Preheat the oven to 350F and line an 8×8" baking pan with parchment paper. Whisk the eggs and sugar with an electric mixer until it's fluffy.
Add the lemon zest and juice, olive oil and vanilla.
In a separate bowl, which the dry ingredients.
fold the dry ingredients into the wet.
Bake for 30 minutes, or until a toothpick comes out clean.
Mix the icing ingredients together and top on the cooled cake
Top with candied lemon slices.