In a large bowl, whisk the brown sugar into the warm water. Once dissolved, sprinkle yeast on top. Let it sit for 10 minutes until it becomes foamy.
Add flour, ½ cup olive oil, salt, and vanilla to the yeast mixture. Mix well until a sticky dough forms. Cover with plastic wrap and let rise for 30 minutes (or up to 2 hours).
To prepare the filling, stir together brown sugar, cinnamon, and melted butter until a spreadable paste forms.
Now, lightly oil a baking pan. Slide dough directly onto the pan. With oiled fingers, stretch the dough into a large rectangle.
Spread about ¾ of the cinnamon filling evenly over the dough.
Fold the dough over itself like a book—left to right, top to bottom—until the filling is enclosed. Gently stretch and press the dough to fit the pan.
Spread the remaining filling evenly over the surface. Let the dough rise for 30 minutes, or until it becomes puffy.
Drizzle the remaining 2 tablespoons of olive oil over the dough. Using oiled fingers, press dimples all over the surface. This helps the cinnamon sugar caramelize.
Preheat the oven to 400°F and bake the focaccia 25–28 minutes, or until golden brown and bubbly.
While the focaccia bakes, prepare the cream cheese icing. Beat softened cream cheese, powdered sugar, vanilla, and milk until smooth. Adjust milk for drizzling consistency.
Once baked, drizzle icing over the warm focaccia and spread evenly. Slice and enjoy warm.