Warm the water to about 105–115°F (warm to the touch). In a large mixing bowl or the bowl of a stand mixer, stir the warm water with the sugar until dissolved. Sprinkle the instant yeast over the surface and let sit 3–5 minutes until foamy. Whisk in the olive oil and the egg until combined.

Mix dough

Add the flour and salt to the bowl. If using a stand mixer, attach the dough hook and mix on low speed until a shaggy dough forms, then increase to medium-low and knead 6–8 minutes until the dough is smooth, elastic, and slightly tacky. If mixing by hand, turn the dough onto a lightly floured surface and knead 8–10 minutes until smooth and elastic. Add up to 2 tablespoons extra water if dough seems too stiff, or up to 2 tablespoons additional flour if excessively sticky.

Bulk rise

Lightly oil a clean bowl with a little olive oil. Place the dough in the bowl, turn once to coat, cover tightly with plastic wrap or a damp kitchen towel, and let rise in a warm draft-free spot until doubled in size, about 60–75 minutes depending on room temperature.

Filling & Shaping

Prepare mix-ins

While the dough rises, pit and roughly chop the kalamata olives, finely chop the basil leaves, and grate the Parmesan. Combine olives, basil, Parmesan, black pepper, and red pepper flakes in a small bowl; set aside.

Deflate and roll

After the dough has doubled, gently punch it down to release air. Turn it out onto a lightly floured surface and press into a rough rectangle about 10x8 inches.

Add filling and form loaf

Evenly sprinkle the olive-basil-Parmesan mixture over the dough, leaving a 1/2-inch border on one long edge. Starting from the opposite long edge, roll the dough tightly into a log, sealing the long seam with gentle pressure and pinching the ends to enclose filling. Place seam-side down into a lightly oiled 9x5-inch loaf pan. If you prefer a free-form loaf, shape into a tight oval and place on a parchment-lined baking sheet seam-side down.

Final rise

Cover the loaf loosely with lightly oiled plastic wrap or a damp towel and let rise until puffy and nearly doubled, about 40–50 minutes. Meanwhile, preheat the oven to 375°F (190°C) for at least 20 minutes before baking.

Baking & Finish

Brush and score

When the loaf has finished its final rise, brush the top with the 2 tablespoons of olive oil. Use a sharp knife or lame to make three shallow diagonal slashes across the top (about 1/4-inch deep). Sprinkle the coarse sea salt evenly over the top.

Bake

Place the loaf in the preheated 375°F (190°C) oven on the middle rack. Bake for 30–36 minutes until the crust is golden brown and the internal temperature reaches 200–205°F (93–96°C). If the top is browning too quickly, tent loosely with aluminum foil after 20 minutes.

Cool and garnish

Remove the loaf from the oven and let rest in the pan for 10 minutes, then transfer to a wire rack to cool for at least 30 minutes before slicing so the crumb sets. Garnish with fresh basil leaves if desired. Slice and serve warm or at room temperature.