To make the dough, combine warm water, 1/2 cup of flour, and yeast to a mixing bowl. Use a whisk to mix everything into a smooth batter. Cover bowl with a towel and let sit for 30 minutes.
Add olive oil, salt, and remaining flour to the bowl, and stir with a spoon or fork until the mixture comes together to form a shaggy ball of dough.
Transfer everything to a work surface and knead until the dough just starts to become elastic, about 2 minutes.
Place dough in a lightly oiled zip-top bag. Press out most of the air, seal, and place in the refrigerator for 8 hours, or overnight.
The next day, take the bag of dough out of the fridge, and leave out at room temperature until it roughly doubles in size, about 45 minutes.
Meanwhile, preheat a charcoal grill until coals are white and ashy. Quarter your peeled and cored pineapple and trim out the tough, fibrous core in the center.
Grill pineapple round-side down over hot coals until heated through, and fairly well charred, about 3 minutes; flip and continue to grill on the cored side until charred, 2 to 3 minutes. Reserve until needed.
Once dough has doubled, remove from bag and divide into 4 portions. Form each portion into a ball and use a rolling pin to roll dough into an oval, or circle about an 1/8-inch thick.
Place dough over hot coals, and grill until dough is cooked through, and about half the surface area is slightly charred, about 2 minutes per side.
Remove from grill and top each warm flatbread with a drizzle of olive oil. Top with prosciutto, thin slices of grilled pineapple, tarragon leaves, and freshly ground black pepper. Finish with another small drizzle of olive oil and a pinch of flaky sea salt.