In a large bowl, dissolve the yeast in 1/4 cup of warm water.
Stir in the sugar/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. (To knead in a food processor, combine the flour, butter and salt in the work bowl of the food processor and add the yeast mixture. With the machine on, gradually add enough warm water until the dough forms a ball, then process around the bowl about 25 times.).
Shape into a ball, place in a large oiled bowl turning 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 spring form 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.
Pre-heat the oven to 375 degrees.
Brush the khachapuri with the egg mixture (Beat the egg with 1 tsp water).
Bake until golden brown, about an hour.
Let cool for 30 minutes before slicing.
Georgian Cheese Filling (Khveli):
In a large bowl, combine all the ingredients and stir to blend.