Combine sugar and yeast in a small bowl. Mix flour, baking powder, xanthan gum, baking soda, and salt together in a separate bowl.

Place milk and butter in a microwave-safe bowl. Cook on high, stirring every 15 seconds, until 110 to 115 degrees F (43 to 46 degrees C), about 45 seconds. Whisk into yeast mixture and set aside to proof, about 5 minutes.

Combine proofed yeast with olive oil, egg, and vanilla extract in a large bowl; mix for a few seconds, then slowly add flour mixture. Mix with an electric mixer on medium-high speed until dough thickens and loses some of its stickiness, about 1 1/2 minutes.

Cover a work surface with plastic wrap and a light layer of flour. Place dough in the center, dust with a little more flour, and cover with another sheet of plastic wrap. Roll out to a 10x13-inch rectangle; remove the top layer of plastic wrap.

Combine brown sugar and cinnamon in a small bowl. Spread melted butter over dough; sprinkle brown sugar mixture evenly over top.

Starting at a long edge, roll dough up like a jelly roll, using the plastic wrap to help lift and roll as you go. Lightly sprinkle dough with more flour and cut into 10 pieces with a sharp knife. Place rolls in a greased baking dish, cover with plastic wrap, and place in a warm spot to rise for 15 minutes.

Preheat the oven to 350 degrees F (175 degrees C).

Bake rolls in the preheated oven until tops are golden brown, 22 to 27 minutes.

Beat butter and cream cheese in a mixing bowl until smooth. Add powdered sugar; mix until smooth. Mix in lemon juice, vanilla extract, and salt.