In the bowl of a stand mixer with the dough hook attachment fitted, combine the yeast, sugar, and warmed dairy-free milk. Stir gently and set aside for 5–10 minutes, or until the mixture becomes foamy.

To the bowl, add the plain flour and salt and begin mixing. Add the yoghurt, melted dairy-free butter, and vanilla extract. Mix on a low speed until the dough comes together. Continue to knead using the mixer for 5 minutes, or until the dough is smooth and elastic. Turn the dough out onto a floured surface and knead again by hand for another 5 minutes to form a smooth and elastic dough. Shape the dough into a ball and place it back into the bowl that has been lightly greased. Cover with a damp tea towel, and let it rise in a warm spot for 1 hour, or until it doubles in size.

Mix the brown sugar and cinnamon in a small bowl for the filling. Once the dough has risen, punch it down and roll it out on a floured surface to a 20x30 cm rectangle. Spread the dairy free butter over the dough, leaving a small border around the edges, and evenly sprinkle over the cinnamon sugar mixture. Using a pizza cutter, slice the dough into six equal strips. Roll each strip into a tight spiral to form individual rolls. Arrange them in a lightly greased baking dish and cover with a damp tea towel. Set aside in a warm place to rise for 30 minutes.

When the rolls have 15 minutes to go in their rise, preheat the oven to 180°C (350°F). Bake the cinnamon rolls for 25–30 minutes, or until golden brown and cooked through. While the rolls are baking, whisk together the icing sugar, dairy-free milk, and vanilla extract to create a smooth icing. Adjust the milk to achieve your desired consistency.

Once the rolls have slightly cooled, drizzle the icing over the top and serve warm.