Make the dough: In the bowl of a stand mixer, whisk together 1 cup (125 grams) flour, sugar, yeast, and salt by hand.
In a small saucepan, heat water, butter, and milk over medium-low heat until butter is melted and an instant-read thermometer registers 120°F (49°C) to 130°F (54°C). Add warm butter mixture to flour mixture. Using the paddle attachment, beat at medium-low speed until combined. Add egg and beat until combined. With mixer on low speed, gradually add 1½ cups (188 grams) flour, beating just until combined and stopping to scrape sides of bowl.
Switch to the dough hook attachment. Beat at low speed until a soft, somewhat sticky dough forms, 5 to 7 minutes, stopping to scrape sides of bowl and dough hook as needed; add up to remaining 2 tablespoons (16 grams) flour, 1 tablespoon (8 grams) at a time, if dough is too sticky. Turn out dough onto a very 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 30 to 45 minutes.
Turn out dough onto a very lightly floured surface, and gently press into an 8x6-inch rectangle. Loosely wrap in plastic wrap. Refrigerate for at least 1 hour or up to overnight on a parchment-lined sheet pan.
Spray an 8½x4½-inch loaf pan with cooking spray. Line pan with parchment paper, letting excess extend over sides of pan.
Make the Harissa filling: In a small bowl, stir and fold together butter, harissa paste, and honey until smooth and combined. Fold in flour, parsley, thyme, and salt. Use immediately.
Make the babka: On a lightly floured surface, roll dough into a 13x10-inch rectangle (about ¼ inch thick). Using a small offset spatula, spread Harissa Filling onto dough, leaving a ½-inch border on both short sides and one long side. Sprinkle goat cheese evenly over filling, make sure there are no large clumps of cheese. Starting with long side opposite border, roll up dough, jelly roll style, and pinch seam to seal. Gently lift log at each end, and stretch to 14 inches long. Place seam side down, making sure seam is off to one side instead of in center. Using a sharp serrated knife, cut in half lengthwise. Turn halves cut side up, and place in an “X.” Twist top half of “X” twice; pinch ends, and tuck under. Repeat with bottom half of “X.” Using your hands, place in prepared pan, making sure to keep cut sides up and ends tucked. Cover and let rise in a warm, draft-free place (75°F/24°C) until puffed and indentation remains after lightly touching top of dough with fingertip, about 30 minutes to 1 hour.
Preheat oven to 375°F (190°C).
Make the egg wash: In a small bowl, whisk together remaining egg and water. Using a pastry brush, brush egg wash on top of dough.
Bake: Bake until golden brown and an instant-read thermometer inserted in center registers 190°F (88°C), 50 to 55 minutes, covering with foil after 20 minutes of baking to prevent excess browning.
Make the Hot Honey Syrup: In a small microwave-safe bowl, stir together all ingredients. Heat on high in 15-second intervals, stirring between each, until honey has softened and syrup is fluid.
Remove babka from oven and place pan on a wire rack. Brush top of loaf with Hot Honey Syrup. Let babka cool in pan for 10 minutes. Using excess parchment as handles, remove from pan, and let cool completely on a wire rack. Garnish with parsley, if desired.