In a food processor, combine flour and salt. Add in butter and pulse until small pebbles form.
In a small bowl, combine sour cream, lemon juice, and water. Add to food processor and pulse until mixture comes together and forms dough. Wrap dough in plastic wrap and refrigerate for at least one hour, or up to two days.
To make the filling: In a heavy bottomed skillet, heat olive oil over medium high heat. Add onions and cook until translucent. Add in garlic, chili pepper, salt, pepper, caraway seeds, cumin, turmeric, and paprika. Cook for 1 minute. Stir in tomatoes, tomato paste, honey, and vinegar. Reduce heat to medium low and simmer for about 10-15 minutes, until it starts to thicken. Add in spinach and mix until slightly wilted. Remove from heat and let cool slightly.
Preheat oven to 400F.
While filling cools, divide dough in 4 even pieces. Roll out each piece of dough into about a 8 inch round. Transfer dough to a parchment paper lined baking sheet.
Place tomato mixture in the center, leaving about a 1½ inch border around the sides. Scatter feta evenly among galettes, then fold the edges of the dough in, overlapping where there is extra.
In a small bowl, whisk one egg with a few drops of water. Brush the dough with egg wash. Bake for 25 minutes, until dough is slightly golden.
Remove galettes from oven and using a spoon, gently make a well in the center of each galette. Crack one egg into each galette, taking care to spread the whites out with a knife so that they don't pool on top of the yolks. If you have big eggs, you may consider using only half of each egg white. You can do this by cracking the egg over a bowl and letting some of the white drip out before putting the yolk and remaining whites into the well in the galette. Bake for an additional 5-8 minutes, until whites have set.
Let cool slightly, then sprinkle with smoked paprika and freshly ground black pepper before serving.