In the bowl of a stand mixer or large bowl, stir together the warm water and yeast. Let sit for a couple minutes until it foams a bit, just to make sure it's not dead.
Meanwhile, in a small bowl melt the 4 tablespoons butter in the microwave. Add the milk and vanilla. Set aside.
Add 2 cups of the flour, the sugar, and salt to the dough and stir together.
Add the eggs and continue to stir on low.
Add in the butter/milk mixture in a thin stream. Add more flour a little bit at a time until the dough comes together in a ball and cleans the sides of the bowl. Knead on medium for 3-5 minutes.
Transfer the dough to a large greased bowl. Cover it with a cloth and let rise in a warm place for and hour and a half. (I like to turn my oven to warm (170F) for about a minute, turn it off, then let the dough rise in the oven.)
While the dough rises, prepare the filling. In a small bowl stir together the brown sugar, cinnamon, and salt. Set aside.
Grease a 9x5 inch loaf pan and set aside.
Toward the end of the dough rising, prepare the peaches. Peel off the skins and chop them into small pieces. Melt 3 tablespoons of butter in a small dish.
Once the dough has risen, turn it out onto a large and well-floured work surface. Roll the dough into a large rectangle, roughly 12x24 inches, but don't fret if it's not exact.) The dough will be pretty thin.
Brush the rectangle with the butter you just melted. Slather it all on, don't be shy. Sprinkle the whole thing with the cinnamon-brown sugar.
Use a pizza cutter to slice the dough into 6 equal strips the LONG way (each strip should be about 24 inches).
Sprinkle the first strip with 1/6 of the chopped peaches (just eyeball it). Now you are going to lay the next strip on top of the one you just sprinkled with peaches. I found it easiest to slide my entire arm underneath the dough, pick it up and lay it gently on top of the peaches.
Sprinkle the stack with more peaches. Add the next layer of dough. Keep going until you run out of peaches and strips of dough.
Now you are going to use a very sharp knife to cut the stack into segments that will fit into your bread pan. I lined up the short end of the bread pan to the edge of the dough, and sliced there. Take the first segment and stack it vertically in the pan, like they are papers in a filing cabinet. Continue until all the dough is in the pan. Don't be afraid to shove it all in there.
Preheat your oven to 350 F.
Cover the dough with a towel and let rise in a warm place for about 30-45 minutes (not in the preheating oven this time.)
Bake for 40-45 minutes until brown and cooked through. Use a toothpick to reach the center of the bread, the layers make it pretty easy to do this. Check after about a half an hour to see if the bread is getting too brown on top. If so, cover with tin foil.
Cool on a wire rack for 10 minutes.
Meanwhile, make the Caramel Sauce. Add the butter, brown sugar, and milk to a small sauce pan and bring to a boil. Boil for 1 minute, then remove from heat.
Turn the bread out onto a cooling rack with parchment or wax paper below it. Drizzle with the warm Caramel Sauce. Save some for dipping if you want. Eat it while it's hot!