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To make the dough: Weigh your flour; or measure it by gently spooning it into a cup, then sweeping off any excess. In the bowl of a stand mixer fitted with the dough hook, mix all the dough ingredients together until a smooth and supple dough forms, 6 to 8 minutes.
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Transfer to a greased bowl, cover, and let it rise until nearly doubled, 1 to 1 1/2 hours. While the dough rises, make the filling.
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To make the filling: Slice off the top of the garlic head to expose the cloves. Place cut side-down in a small microwave-safe bowl. Add the olive oil and water. Microwave, uncovered, for 4 to 6 minutes at 50% power, until the garlic cloves are soft. When cool enough to handle, squeeze out 4 to 6 garlic cloves (20g) into a medium bowl. (Save the rest of the garlic for another use or add more if you’re a garlic lover!)
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Mash the cloves with a fork until smooth. Add the cream cheese, Parmesan, herbes de Provence, and salt; mix to combine.
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To shape the tarte soleil: Divide the dough in half (335g per piece), shape each piece into a tight ball, cover, and let them rest for 10 minutes.
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On a lightly floured surface, roll one ball into a 10 1/2" circle. Transfer to a sheet of parchment, then roll out the other ball to the same size.
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Spread the filling over the dough on the parchment, leaving a bare 1/2" border. Brush the border with egg wash. Place the bare circle of dough on top of the filling; gently press to seal the borders together.
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Gently place a 2 1/2" to 3" round cutter in the center of the dough and use a sharp knife or pizza wheel to cut the circle into 16 equal strips, from the cutter to the edge. It helps to work in quadrants: Cut the dough into 4 sections, then cut each section in half, then cut each of those sections in half again.
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Remove the cutter from the center of the dough, and transfer the dough, parchment and all, to a baking sheet. Next, gently tug each strip away from the center, then twist 3 or 4 times in the same direction.
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Cover the dough and let it rise for 45 to 60 minutes, until it's puffy and bounces back slowly when poked.
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Near the end of the rising time, preheat the oven to 400°F with a rack in the center.
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Brush the dough all over with egg wash. In a small bowl, mix together the Parmesan and the herbes de Provence, then sprinkle this mixture around the edge of the bread and in the center.
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Bake the tarte soleil for 20 to 25 minutes, until deeply golden. Let cool slightly, then serve warm or at room temperature.
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Storage information: Wrap any leftover tarte soleil airtight and store at room temperature for up to 2 days. To reheat, wrap the tarte soleil in foil and bake in a 325°F oven until warm, 10 to 15 minutes.
