Add the strawberries, sugar, cornstarch or arrowroot, and lemon zest to a small pan and cook down over a medium heat stirring frequently until jammy and slightly thickened. It only takes around 5 minutes. Remove from the heat and set aside to cool.
For the dough
Add the flour, yeast, salt, and sugar to the bowl of a stand mixer. If you don't have a stand mixer you can mix and knead the dough by hand.
Next add the dairy-free milk, butter (be sure the melted butter has cooled to room temperature), and vanilla.
Mix with the dough hook until combined. If you need to add a little extra milk. The amount needed will vary depending on the brand of flour and humidity. The dough should be soft but not too sticky as it needs to be roll-able.
Continue kneading in the machine for about another 5 minutes until soft and smooth. If you are kneading by hand you will need to knead for at least 10 minutes.
Once smooth and supple, transfer to a greased bowl, cover with a damp dish towel and wait until almost doubled in size. The time this takes will vary dependent on temperature. In my 21°C (70°F) kitchen it usually takes about 40 minutes.
Once it's well-risen turn the dough out onto a very lightly floured work surface and roll out into a 9 x 14 inch rectangle.
Spread the cooled strawberry filling evenly over the dough leaving a 1-inch margin around the edges, then roll up tightly starting along the long edge.
Slice the roll up into 8 even pieces. I find a very sharp serrated knife best for this.
Place each roll into a vegan butter-greased dish or pan. I use an 8 x 12 pan but a large pie dish will do or anything similar. Cover with the damp dish towel again and place them on the stove top or near the oven to keep them warm. Leave for 30 minutes.
Preheat oven to 350°F (175°C).
Bake for anything from 22 to 30 minutes. At 22 minutes they should be cooked through. If you want them super soft and squidgy remove now. If you like them slightly crusty on the outside they will take around 30 minutes.
Once done place the pan on a wire rack and allow to cool a little.
To make the icing, add the lemon juice very gradually to the powdered sugar, stirring as you go. Start with about 1½ tablespoons, stir well, then keep stirring while adding a small drop or two until it's very thick but drizzle-able. At first it won't seem enough liquid for all of that sugar but it will be plenty. You might not need all of the lemon juice.
You can drizzle the icing over the buns once they've cooled for about 10 minutes for a sticky glaze-type effect, or if you like visible swathes of thick white icing, let them cool completely first. Once iced a sprinkling of lemon zest is optional but looks really pretty!
NOTES
Store in an airtight container for 2 - 3 days. Reheat in a microwave for 15 - 20 seconds to make them really soft and almost like freshly baked again.