Heat the milk, in the microwave or a saucepan, until it is “fingervarm” (finger warm), meaning it should feel warm to the touch but not hot. Add your yeast and 1 tbsp of sugar, stir, and leave it for 5-10 minutes until the mixture starts to bubble up, meaning your yeast is alive and not expired.
Add the remaining sugar (2 tbsp), egg, carrots, sunflower seeds, and salt to a large bowl and mix well. Once the yeast mixture has bubbled up a bit, add it to the egg mixture. Mix well.
Slowly add the flour and softened butter, mixing with a wooden spoon until the dough starts to come together.
Knead until smooth and elastic, about 10 minutes. Add more flour if the dough is too sticky; it should be soft and smooth, not too sticky nor too dry. Use a stand mixer if you would prefer!
Cover the dough with cling film or a clean dishtowel and place in the fridge overnight for 8 hours, or let rise at room temperature (if room temp is very cold wherever you are, put it in a warm place, i.e. under a light) for 1.5-2 hours or until doubled in size.
Once your dough has doubled in size, shape it into about 15 balls and place on two baking sheets lined with parchment paper. You can also crowd them all onto one pan if you would prefer buns that stick together as they bake. To shape the rolls properly, you should first divide your dough into equal sections. Take a piece, pinch together into a ball shape, then cup it in your hand with the top of the roll touching your palm. Place your hand on an unfloured surface with the seam touching the table. Move your cupped hand in quick, tight circles – this will cause the seam to come together and create tension on the surface of the dough, meaning it will rise upwards in the oven, instead of flattening.
Cover and let rise for another 30 minutes. In the meantime, preheat your oven to 400 degrees F (205 degrees C).
Brush with egg wash (whisk together one egg and some water until there are no more gloopy egg streaks). Bake for 10-12 minutes. Check the bottom of the buns to see whether they are done – they should be golden brown. If they’re too pale, they’ll need another minute or two. Enjoy!