In a small bowl, combine the hot water, sugar, and instant yeast. Stir gently and let sit for 5 minutes until the mixture becomes foamy and bubbly.

In a large mixing bowl, whisk together the flour, baking powder, and salt. Make a well in the center and add 3 tablespoons of olive oil, the sparkling water, and the yeast mixture. Stir until just combined; the dough will be very sticky and lumpy.

Cover the bowl tightly with plastic wrap and then a clean kitchen towel. Let the dough rise in a warm, draft-free spot until doubled in size, about 2 to 4 hours.

Butter a 9x13-inch baking dish with the softened butter. Transfer the risen dough to the dish (it will deflate). Gently pull and stretch the dough to the edges. Fold the top edge down toward the center, then fold the bottom edge up. Rotate the dish and press the dough lightly with your fingertips to dimple it. Cover again with plastic wrap and let rise until more than doubled, about 1 to 2 hours.

Preheat your oven to 425°F. Drizzle the remaining 2 tablespoons of olive oil over the top of the dough. Use your fingertips to press dimples all over the surface, creating air pockets. Bake for 25 to 30 minutes, until deep golden brown. Remove from the oven and let cool in the pan for about 10 minutes.

Spoon the peach jam into a piping bag fitted with a round tip. Poke several holes into the top of the warm focaccia and pipe the jam into each hole, filling generously.

In a medium bowl, whisk together the powdered sugar, cream extract, and milk until smooth and pourable. Drizzle the icing over the filled focaccia.

Slice into squares and serve warm or at room temperature. Enjoy your jelly doughnut focaccia!