In the bowl of a stand mixer, whisk together 2 cups (250g) flour, sugar, yeast, and salt by hand.
In a medium saucepan, heat water, milk, and butter over medium heat, stirring occasionally, until butter is melted and an instant-read thermometer registers 120°F (49°C) to 130°F (54°C). With mixer on medium speed and using the paddle attachment, pour warm milk mixture into flour mixture, beating until combined and cooled slightly, about 1 minute. Add 1 egg (50g) and beat at medium speed until combined. Reduce mixer speed to low, and gradually add 1 3/4 cups (219g) flour, beating until a shaggy dough forms. Switch to the dough hook attachment, and beat at low speed until dough is smooth and elastic, 12 to 14 minutes. Add remaining 1 tablespoon (8g) flour if dough is too sticky. Turn out dough onto a lightly floured surface, and shape into a smooth round.
Lightly oil a large bowl. Place dough in bowl, turning to grease top. Cover and let rise in a warm, draft-free place (75°F/24°C) until doubled in size, about 45 minutes.
Preheat oven to 375°F (190°C). Line 2 baking sheets with parchment paper.
Make Feta Cheese filling: In the bowl of a stand mixer fitted with the paddle attachment, beat all ingredients at medium-low speed until well combined and creamy, about 1 minute. Set aside.
Punch down dough, and let stand for 5 minutes. Turn out onto a lightly floured surface, and divided into 12 portions (70g each). Shape each portion into a smooth ball; press flat, and roll into a 3-inch circle. Place 6 circles 3 inches apart on each prepared pan. Cover and let rise in a warm, draft-free place (75°F/24°C) until puffed, about 15 to 25 minutes. Press bottom of a 4-ounce glass Mason jar (2-inch diameter) dipped in flour into center of each circle. Press firmly to flatten center without tearing dough. Spoon 2 rounded tablespoons (about 36g) Feta Cheese Filling into center of each circle. (Do not overfill.)
In a small bowl, whisk together remaining 1 egg (50g) and remaining 1 tablespoon (15g) water. Lightly brush dough with egg wash.
Bake until light golden brown and an instant-read thermometer inserted in bread registers 190°F (88°C) to 200°F (190°C), 12 to 13 minutes. Let cool on pans for 5 minutes. Serve warm or at room temperature. Garnish with tarragon and thyme, if desired.