Whisk lemon sugar and eggs on high for 4-5 minutes, until thick. On medium high speed, slowly stream in olive oil. Beat on high an additional minute. On low speed, stream in milk. Beat on low to fully combine. Fold into the dry ingredients.
Pour the lemon juice and sugar into a small sauce pan. Heat over medium heat unil the sugar fully dissolves. This takes about 5-10 minutes once the liquid begins to simmer. Remove from heat and pour into a heat proof bowl or cup.