Preheat oven to 350F.
In a large bowl, add melted butter and sugar. Mix until combined, then add eggs one at a time, mixing in between. Add milk, greek yogurt and vanilla. Mix until batter is smooth.
Add dry ingredients to wet and fold to combine. Place batter in a lined 8x8 baking dish and bake for 30-32 minutes.
Allow cake to cool fully before icing.
For blackberry jam, add blackberries, sugar, lemon juice and vanilla to a small saucepan on medium heat. Cook until all liquid has evaporated, about 12-15 minutes, stirring often. Allow to cool fully before making icing.
For icing, add butter to a stand mixer and mix on high for 2 minutes. Add powdered sugar, vanilla and milk and mix on high again for 5 minutes. Add cooled blackberry jam and mix until just incorporated.
Top cooled cake with frosting and enjoy!
