In a stand mixer bowl, combine the water, active sourdough starter, and raw honey. Mix on low for just a few seconds until the starter is mostly dispersed; a few small lumps are totally fine.

Add the bread flour and coarse sea salt. Knead on low speed for 5 to 7 minutes. The dough will be sticky and wet at first, then pull away from the sides of the bowl and cling to the hook.

Cover the bowl and let it rest for 30 minutes.

With lightly wet hands, grab one side of the dough, stretch it upward, and fold it over the center. Rotate the bowl 90 degrees and repeat. Complete 4 stretches to make one set, then cover and rest for 30 minutes.

Repeat the stretch and fold process for 3 sets total, with a 30 minute rest after each set. After the last set, brush a large clean bowl with a little butter and transfer the dough into it for bulk fermentation.

Keep the dough in a warm spot until it has nearly doubled, about 3 to 4 hours depending on your kitchen temperature. While it rises, make the apple fritter filling.

Melt the butter in a small saucepan over medium-low heat. Whisk in the granulated sugar and bread flour and cook until the mixture thickens and bubbles.

Peel and dice the apple, then stir it into the saucepan with the brown sugar, cinnamon, and coarse sea salt. Cook over medium-low heat until the apples soften slightly and the mixture thickens. Remove from the heat and cool completely before shaping.

When bulk fermentation is complete, pour 2 tablespoons of the melted butter into a greased 9-by-13-inch pan. Drizzle the remaining 1 tablespoon of melted butter over the dough.

Use a bowl scraper to turn the dough out into the buttered pan. Gently press it into a rough rectangle, then spread half of the apple filling over the surface.

Fold in the long sides, then the short sides, to make a rectangular envelope. Flip the dough seam-side down. Spread the remaining filling on top, cover the pan with plastic wrap, and proof until nearly doubled, about 2 to 3 hours. You can bake right away or refrigerate overnight for up to 2 days.

Melt the remaining butter. Coat your fingers with a little of it and drizzle the rest over the dough. Use your fingertips to dimple the surface firmly, pressing in and wiggling slightly. Stretch the dough gently to fill the pan if needed.

Preheat the oven to 425°F and let the dough rest while the oven comes to temperature. Bake for 25 to 30 minutes, until golden brown and an instant-read thermometer inserted into the center reads about 200°F.

Cool the focaccia in the pan for at least 10 minutes, then transfer it to a wire rack. While it cools, whisk together the powdered sugar, milk or half-and-half, vanilla paste, and a pinch of coarse sea salt to make a smooth glaze.

Drizzle the glaze evenly over the warm loaf and serve right away.