Mix all dry ingredients until combined. Set aside.
In a glass measuring cup, add water and butter and microwave until just melted. In small bowl combine ground flax seed and hot water.
Stir in wet ingredients and flax seed mixture with the dry.
Slighty dampen your counter top and put down a piece of plastic wrap so that it covers an area bigger than 8"x16". To avoid sticky dough, spray some oil onto the plastic wrap, than sprinkle with about 2 tablespoon of sugar. Lay ball of dough on plastic wrap and cover with another piece of plastic wrap. Roll the dough out to approximately 10"x18". Remove the plastic wrap.
Mix cream cheese, brown sugar, and cinnamon together in a medium mixing bowl until thoroughly combined.
With a spatula, spread the filling mixture evenly onto dough, leaving an inch around the edge without any filling. Evenly sprinkle pecans over filling.
Prepare a baking sheet with parchment paper or a Silpat and set right next to your dough. Carefully along the long end, start rolling it up into a long cylinder. Using the plastic wrap, lift the roll onto your baking sheet and placing it on its side. Remove the plastic wrap and gently form the roll into a circle and pinch the ends together.
Whisk egg wash ingredients together. Using a pastry brush, brush top of king cake. Cover loosely with plastic wrap and let it rise in a warm place for 30-40 minutes until nearly doubled in size.
Preheat oven to 350. Bake for 30-35 minutes.
Take out of oven and let sit for 10 minutes. Using two large turners, move to cooling rack.
Mix icing ingredients together and drizzle icing onto top of king cake. When cool, cover top of king cake in thirds with purple, yellow, and green sprinkles.