Gel the psyllium. Whisk the psyllium husk powder into the lukewarm water and let it sit for about 10 minutes, until thick and gel-like.
Bloom the yeast. Pour the warm milk into a large mixing bowl and stir in the granulated sugar. Sprinkle the instant yeast over the top, cover with a kitchen towel, and let it bubble up for 1–2 minutes. (If the milk is too hot it will kill the yeast — it should feel like warm bathwater, 110°–115°F.)
Build the wet base. Whisk the room-temperature eggs in a small bowl (bring whole eggs to room temp first by sitting them in warm water for 5 minutes). Add the whisked eggs and the apple cider vinegar to the milk mixture and stir to combine. Stir in about two-thirds of the psyllium gel.
Mix the dough. In the bowl of a stand mixer fitted with the paddle attachment, whisk together the flour blend, baking powder, and salt. Add the dry mixture to the wet base along with the melted butter and mix 1–2 minutes, scraping down the sides, until fully combined.
Switch to the dough hook and mix on medium speed for 3–5 minutes. The dough will start to pull away from the sides of the bowl. It should be tacky and soft — do not add more flour. Gluten-free yeast dough is softer than traditional yeast dough. If it seems stiff rather than tacky, beat in the remaining psyllium gel 1 tablespoon at a time.
First rise. Transfer to a greased, heat-proof bowl. Wet your hands with warm water and rub the dough ball smooth (this adds moisture for the rise). Cover with plastic wrap. Set the bowl inside a larger heat-proof bowl, pour 2 cups of microwaved-hot water into the larger bowl (none into the dough bowl), cover both with a kitchen towel, and let rise 20 minutes.
Make the filling. In a small bowl, mix the softened 1/4 cup butter with the vanilla until smooth. In another small bowl, mix the brown sugar and cinnamon.
Shape. Spray a piece of parchment paper with gluten-free cooking spray. Place the risen dough on it, dust the top with 1 tablespoon of gluten-free flour, and roll to a 12×14-inch rectangle about 1/2-inch thick, dusting the rolling pin as needed. Spread the butter mixture edge to edge, sprinkle over the brown sugar mixture, and pat down with your hands.
Using the edge of the parchment, fold the long edge up and roll the dough into a tight log. Cut the log in half, then cut each half into 6 equal rolls — unflavored dental floss gives the cleanest cuts.
Second rise and bake. Arrange the rolls in a greased 9×13-inch baking dish. Warm the heavy cream (30 seconds in the microwave) and pour it over the rolls. Place the dish on the middle oven rack and turn the oven to 375°F — the oven coming up to temperature is the second rise. Once it reaches 375°F, bake 15–20 minutes until light golden brown; check at the 15-minute mark. The center of the centermost roll should read about 200°F on an instant-read thermometer so the psyllium-moistened centers aren't gummy.
Frost. Beat the cream cheese frosting ingredients (or stir the vanilla icing ingredients) until smooth, and spread over the warm rolls. Cool a few minutes before serving.
Cover the shaped, unbaked rolls with plastic wrap and refrigerate overnight. Bring to room temperature, then place on the middle rack, turn the oven to 375°F, and bake 15–20 minutes once it reaches temperature.
Freeze after cutting, pan covered in foil. Thaw overnight in the refrigerator, bring to room temperature, then bake as above.
Cool completely, wrap in foil, and freeze in a bag or container. Defrost to room temperature and warm at 350°F for 10–15 minutes (or microwave individual rolls 20–30 seconds). Store leftovers airtight; reheat at 350°F for 10–15 minutes or microwave 20–30 seconds.