In the bowl of a stand mixer fitted with the dough hook, combine the flour, eggs, milk, sugar, yeast, and butter. Mix on low speed until a shaggy dough forms, then knead until smooth and elastic, about 10 to 12 minutes. Add the salt and knead until fully incorporated, about 2 minutes more.
Transfer the dough to a lightly greased bowl, cover, and let rise at room temperature until doubled in size, 1 to 1½ hours.
Meanwhile, make the herb oil. Stir together the dill, parsley, garlic, salt, and olive oil in a small bowl. Set aside.
Line 2 baking sheets with parchment paper. Turn the dough onto a lightly floured surface and divide into 14 equal portions, about 70 grams each. Shape into tight rounds and arrange on the prepared baking sheets, leaving space between each bun. Cover loosely and let rise until puffed and slightly wobbly, 30 to 45 minutes.
Heat the oven to 350°F /180°C. Place a shallow pan of hot water on the bottom oven rack.
Whisk together the egg, date syrup, and water. Brush the buns evenly with the egg wash.
Bake until deeply golden brown, 18 to 20 minutes. Immediately brush with the herb oil and let cool slightly before serving.