Preheat the oven to 350°F. Line bottom of 13"x 9" rectangular baking pan with parchment paper and grease with 1 Tbsp olive oil.
In large mixing bowl, whisk together semolina, baking powder, baking soda, salt, and sugar. Add eggs and buttermilk to dry ingredients and whisk until combined. Add olive oil, lemon zest, and vanilla to batter and whisk until smooth. In separate bowl, toss apricots with 1/2 cup of sugar and arrange halves cut side down over parchment. Pour batter into prepared cake pan over apricots. Bake in preheated oven 30-35 minutes. Take care not to overbake as this cake can dry out quickly.
While cake is baking, make apricot balsamic syrup by placing honey and apricot balsamic in saucepan. Bring to bare simmer over medium heat and reduce 10 minutes until thickened and reduced by 1/3. When cake is finished baking, remove from oven and slowly pour syrup over top of cake until all syrup soaks in. Let cool. Serve at room temperature with a dollop of Greek style yogurt.