For the dough: line a springform pan with butter. In a large bowl, dissolve the yeast in 1/4 cup of warm water. Stir in the honey and let stand until foamy, 5 to 10 minutes. Add the remaining water, the butter and the salt.
Stir in 2 cups of the flour. Gradually add enough of the remaining flour so that the mixture holds together.
On a floured surface, knead the dough until smooth and elastic, about 10 minutes. Shape into a ball, place in a large oiled bowl, turn to coat, cover and let rise until nearly doubled in bulk, about 1-1/2 hours.
Punch down the dough and roll into a 20-inch round about 1/4 inch thick. Fit the dough into the greased springform pan, letting the excess drape over the sides.
Spoon in the filling. (Filling directions at the end of the recipe). Gather the edges together to meet in the center, pleating the dough as evenly as possible into overlapping folds. Twist the dough ends that meet in the center to form a knob.
Cover and let stand for about 30 minutes. Brush the khachapuri with the egg mixture (beat the egg with 1 tsp. water).
Bake in a preheated 375-degree oven for about 1 hour, until golden brown.
Let cool for 30 minutes before slicing.
For the filling: in a large bowl, combine all the ingredients and stir to blend.