Put dates in a bowl with boiling water to soften for about 10-15 minutes.
Preheat oven to 350 F.
Melt butter in a heavy bottomed saucepan on very low cautious not to burn it.
Remove from heat as soon as liquid, stir in honey and wait for it to cool down to room temperature.
Once at room temperature mix in the egg. (Be patient, and wait for the butter to really be at room temperature. If the butter is too hot when you mix in the egg, you will have pieces of fried egg in your muffins.)
Mix flour, baking powder and grated carrots in a separate bowl with a dry whisk.
Once well combined add butter honey egg mixture to dry ingredients and combine well.
Fill silicone mini muffin forms about 4/5 full and bake for 14-16 minutes. (Insert a wooden stick into the batter. When it comes out clean, the batter is ready.)
While muffins are in the oven prepare frosting.
Drain dates until all water is gone.
Add dates, walnuts and carrot juice (preferable freshly made in a juicer) to blender and blend until smooth. (Depending on the quality of your blender you may have to scrape down the sides with a rubber spatula several times. (My blender is very weak and I had to do this like 20 times). Make sure to disconnect your blender every time you insert your spatula to avoid accidents.)
Cover baked and completely cooled (!) muffins with frosting and refrigerate for about 30 minutes.