In the bowl of a stand mixer fitted with a dough hook, combine the warm water, yeast and 1 tablespoon sugar. Let stand until foamy, about 5 minutes.
Add flour, salt, egg (if making non-vegan version), 3 tablesppons sugar and 60 grams (4 tablespoons) melted butter in the stand mixer bowl and mix at low speed for about 5 - 7 minutes, depending on room humidity. Make sure to pause the mixer a couple of times to scrape the sides and bottom. Dough should be smooth and will bounce back when pressed with a finger; the sides of the bowl will be almost clean. Cover the bowl with a plastic wrap or clean kitchen towel and place in a warm, draft-free area until it doubles in size, about half an hour. An oven that has not been turned on will be a good spot for this.
Once doubled in size, gently punch down the dough and scrape the sides and bottom. Cover again and let it rise again for another 30 minutes.
Pre-heat the oven to 400°F. Using a pastry brush, spread 15 grams (1 tablespoon) of melted butter on the rimmed baking sheet. Transfer the dough onto the prepared sheet and use your fingers to press and stretch the dough into a large oval or rectangle, about 10-by-15 inches. Leave the dough to rise in a warm spot, uncovered on the pan, for 30 minutes.
In the meantime, toss the peaches with the remaining 2 tablespoons sugar and the thyme leaves.
Use your fingers to make indentations on the dough for placing the peach slices. Gently push each peach slice on the dough, and avoiding any extra liquid from the fruit in the bowl. Sprinkle demerara sugar evenly on top, then bake for 20 minutes until it's puffed and starts to turn golden brown. , puffed and set, 20 to 25 minutes.
Take out of the oven and brush the remaining 30 grams (2 tablespoons) of melted butter over the bread and fruit. Place in the oven and bake for an additional 5 minutes.
Cool in the pan on a wire rack for 10 minutes before transferring into a cutting board for slicing. Serve warm.