In a medium saucepan, heat milk and ¼ cup of sugar over medium heat, stirring frequently until simmering. Turn off the heat and add butter to the hot milk, stir until completely dissolved. Allow the mixture to cool.
In the meantime, combine warm water, 1 teaspoon of sugar and 1 teaspoon of active dry yeast in a small bowl and let the yeast bloom for 5 - 10 minutes.
Add flour, salt, and cinnamon to the stand mixer bowl and stir to distribute evenly. Toss in the chopped dried figs.
When the milk mixture is cooled to below 110°F and the yeast is bubbly, add them to the flour mixture. Knead with the dough hook for just a few minutes until the dough forms.
Scrape the dough out onto a floured countertop and shape it into a ball. The dough will be soft and somewhat sticky, try not to add too much extra flour, just enough to keep it from sticking to your hands and the counter. Place the dough ball into an oiled glass bowl, cover with plastic wrap and allow to rise for about 1 - 2 hours, until at least double in size.
Oil or butter a 9”x13” baking dish and set aside.
When the dough is ready, dump it out onto a floured countertop and divide it into 12 equal portions. Knead and shape each portion into a small ball, place dough balls into the prepared baking pan, and allow to rise the second time for 30 minutes to an hour.
Preheat the oven to 400°F.
Beat an egg with a tablespoon of whole milk to make egg wash. Brush egg wash on top of the risen buns generously.
Mix flour and 5 - 6 tablespoon of water in a small bowl until a soft pipeable paste forms. Add the paste to a piping bag or a zip top bag and snip the corner to create a small hole. Pipe the flour paste on top of the buns in a cross pattern.
Bake for 20 minutes until the top is golden brown. Allow to cool to the touch on a wire rack. Best served slightly warm.