In a liquid measuring cup, combine the buttermilk, yeast, and sugar.
Mix gently and let sit for 10 minutes until foamy.
While the yeast is bubbling, combine 1/2 cup of sugar with 1/2 teaspoon of dried lavender in a coffee grinder. Grind until it reaches a fine powder.
Soften the butter and whisk it into the measuring cup with the yeast, followed by the eggs.
In a mixer with a dough hook attached (or in a bowl with a wooden spoon), combine all wet ingredients with salt and begin to slowly incorporate flour.
Dough is ready to rest when it has all come together, is smooth, and only slightly sticky to the touch.
Let rest 1 hour.
While the dough is resting, pulse an additional 3/4 cup sugar with an additional 1 teaspoon dried lavender in the coffee grinder until fine.
Mix this with 4 tablespoons melted butter and set aside, along with 1/4 cup turbinado sugar.
Grease two 9" round tins with butter.
When the dough has rested, roll it out on a very lightly floured surface into a 12" x 18" rectangle.
Spread gently with the butter/sugar filling and sprinkle the turbinado evenly on top.
Roll up tightly on the long side to create a log.
Cut in to 1" rounds and arrange carefully in the baking tins.
Let rest for 30 minutes.
Preheat the oven to 350F.
Tent each pan with tin foil and bake for 25 minutes.
Remove the tent and bake for an additional 20-30 minutes, or until the rolls are evenly browned and the center dough is fully baked.
While the rolls are baking, combine all the icing ingredients in a bowl and whisk well.
When the rolls are done, let them cool for about 5 minutes prior to icing.
Ice and serve immediately!