In a large mixing bowl, combine the bread flour, garlic powder, salt, and active dry yeast. Make a well in the center and add the olive oil and squid ink. Slowly pour in the warm water while mixing with a wooden spoon or your hands until it forms a sticky dough. Knead the dough on a floured surface for about 8-10 minutes until it's smooth and elastic. Place it in a greased bowl, cover it with a clean kitchen towel, and let it rise in a warm spot for about an hour or until doubled in size.
While your dough is rising, heat butter in a skillet over medium heat. Add the thinly sliced shallots (or red onion) and cook until they are soft and translucent, about 3-4 minutes. Toss in the sliced garlic and continue to sauté for another minute until fragrant. Now add the button mushrooms and cooking apple juice, stirring occasionally until they're browned and tender. Season with fresh thyme leaves, salt, and pepper to taste.
Preheat your oven to 475 degrees Fahrenheit (245 degrees Celsius). If you have a pizza stone, place it inside while preheating.
Once your dough has risen, punch it down to release any air bubbles. Divide the dough into four equal portions. On a lightly floured surface, roll each portion into a circle or oval shape about ¼ inch thick.
Spread a generous layer of jarred pesto sauce over each flatbread base. Then evenly distribute the sautéed mushrooms over each one along with any additional fresh herbs you fancy.
Carefully transfer each flatbread onto your preheated pizza stone or baking sheet lined with parchment paper. Bake them in the oven for around 10-12 minutes or until they are golden brown and crispy around the edges.
Once out of the oven, drizzle balsamic vinegar mixed with honey over your flatbreads. Slice them into wedges and serve.