Line bottom of a 13×9-in. (3 1/2-qt.) casserole slow cooker or a 6-qt. oval slow cooker with parchment paper, extending paper halfway up sides. (Or use a slow-cooker liner.) In a food processor, add flour, baking powder, salt, and baking soda. Pulse to combine. Pulse in the cubed butter and half-and-half.
Turn dough out onto a lightly floured surface. Knead gently for ten to 15 strokes or until smooth. Roll dough into a 15×10-in. rectangle; spread with the 1/3 cup softened butter. In a small bowl, combine the sugars, cornstarch, and cinnamon. Sprinkle over dough. Roll up rectangle, starting from a short side; seal seam. Slice into ten 1-in. pieces.
Arrange rolls, cut sides down, in prepared cooker. Sprinkle with any excess sugar mixture. Cover and cook on high for 1 3/4 to 2 hours or until a thermometer registers about 185°F, giving crockery liner a half turn halfway through, if possible. (Tops of rolls will be slightly wet.)
If possible, remove the slow-cooker insert. Cool rolls, uncovered, 30 minutes. Spread or drizzle with Cream Cheese Icing.
In a medium bowl, stir together 2 oz. softened cream cheese, 1 Tbs. butter, and 1/2 tsp. vanilla. Stir in 1/2 to 1 cup powdered sugar and a few drops of milk, as needed, to reach spreading consistency.