Prepare the dough by combining the water, honey, olive oil, and yeast in a mixing bowl. Let rest for five minutes or until the mixture is foamy.
Add in the flour and salt, and mix until well-combined, but still sticky. Remove from bowl, and knead for 5-7 minutes, or until the dough is stretchy and smooth. You may also do this in a stand mixer with a dough hook attached. Mix on a medium speed until mixed, then knead on low speed for 5-7 minutes until the dough forms a smooth and stretchy ball on the hook.
Transfer the dough to an oiled bowl, cover, and let rise in a warm place for about a hour, or until noticeably puffy.
To make the flatbreads, heat the butter in a heavy-bottom saucepan over medium-high heat. Add in the sliced onions, and cook until just translucent, about three minutes.
Reduce the heat to low, and cook, stirring occasionally, for about 25-30 minutes, or until the onions are dark, sticky, and sweet-tasting. Remove from heat. Meanwhile, preheat the grill to high and oil the grill grates.
To assemble the flatbreads, divide the dough into thirds, and then stretch one of the thirds into a flat oval shape.
Using a pastry brush, coat one side of the flatbread, and then place the dough, oil-side down, on the grill. Close the grill lid, and cook for 2-3 minutes, or until just puffy.
Brush the top with more oil, and then flip. Top with 1/3 of the apple butter, followed by 1/3 of the onions, 1/3 of the blue cheese and 1/3 of the mozzarella.
Close the grill lid and cook just until the cheese is melted, about 2 minutes. Top with fresh cracked pepper.
Repeat with remaining dough and toppings. Slice into four pieces before serving.