Combine sugar and yeast for dough in a small bowl. Mix 1 3/4 cups flour, baking powder, xanthan gum, baking soda, and salt together in a second bowl.

Place milk and butter in a microwave-safe bowl and microwave on high, stirring every 15 seconds, until 110 to 115 degrees F (43 to 46 degrees C), about 45 seconds. Whisk into the 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 moment and then slowly add in the 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 a sheet of good quality plastic wrap and a light layer of flour. Place dough in the center and cover with a little more flour and another sheet of plastic wrap. Roll out to a 10x13-inch rectangle; remove top layer of plastic wrap.

Combine brown sugar and cinnamon for filling in a small bowl. Spread melted butter over the entire dough rectangle and sprinkle brown sugar mixture evenly over top.

Start at a long edge and roll the dough, 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 pan, cover with plastic wrap, and place in a warm spot to rise for 15 minutes.

Meanwhile, 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.

While rolls are baking, beat butter and cream cheese for frosting in a mixing bowl until smooth. Add powdered sugar; mix until smooth. Mix in lemon juice, vanilla extract, and salt.

Remove cinnamon rolls from the oven and immediately drizzle frosting over top.