Heat the water, milk, and butter in a small saucepan until the mixture reaches 110°–115°F. Pour it into the dry ingredients along with the egg.

Using the dough hook, mix on low speed until the dough comes together. Add flour a tablespoon at a time as needed until the dough is smooth, elastic, and slightly tacky but not sticky. Knead with the mixer for 5–6 minutes.

Transfer the dough to a lightly greased bowl, cover, and let rise until doubled in size, about 1 hour.

Once risen, place the dough on a floured surface and roll into a large rectangle. Spread with softened butter, then sprinkle with the cinnamon, brown sugar, and nutmeg. Scatter the diced apples evenly over the top.

Slice the dough into strips with a dough cutter, then roll each strip into a coil. Place the rolls in a greased 9×13-inch baking dish, leaving a little space between each. Cover and let rise until puffy. Preheat the oven to 350°F.

While the rolls cool, melt the caramel bits and heavy cream in a medium saucepan over low heat, stirring often until smooth. Pour the caramel sauce over the cooled rolls.