For dough: In the bowl of a stand mixer, whisk together 1 cup (125 grams) flour, granulated sugar, yeast, and salt by hand.

In a small saucepan, heat ⅓ cup (80 grams) water, milk, and butter 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 milk mixture to flour mixture; using the paddle attachment, beat at medium-low speed until combined. Add egg, beating until combined. With mixer on low speed, gradually add 1½ cups (188 grams) flour, beating just until combined and stopping to scrape paddle and bottom and sides of bowl.

Switch to the dough hook attachment. Beat at low speed until a soft, somewhat sticky dough forms, 8 to 10 minutes; 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 spray a large bowl with cooking spray. 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, 30 to 45 minutes.

For filling: In a medium bowl, stir together melted chocolate and butter until combined; stir in confectioners’ sugar, cocoa, and salt until smooth.

Line rimmed baking sheets with parchment paper.

Punch down dough, and divide in half (about 314 grams each). On a lightly floured surface, roll half of dough into a 13x11-inch rectangle, with one long side closest to you. (Keep remaining dough covered until ready to use.) Spread half of filling (about ¼ cup or 90 grams) in an even layer onto dough, leaving a ½-inch border on all sides.

Fold dough in half lengthwise, pressing edges to seal. Gently press dough into a 13x7-inch rectangle, making sure edges are sealed. Trim ¼ inch off sealed ends. Cut dough into 5 (2½-inch-wide) rectangles. Cut each rectangle diagonally into 2 triangles. Gently stretch 1 triangle to even out the length of the two sides. Starting from base, roll up triangle toward point. (Do not roll too tightly, or it will fall over while proofing or baking.) Repeat with remaining triangles, placing 2 inches apart on prepared pans. Repeat procedure with remaining dough and remaining filling. Cover and let rise in a warm, draft-free place (75°F/24°C) until puffed, 30 to 45 minutes.

Preheat oven to 375°F (190°C).

In a small bowl, whisk together egg and 1 tablespoon (15 grams) water; gently brush onto dough.

Bake until golden brown, 12 to 15 minutes, rotating pans halfway through baking.

Meanwhile, in a small microwave-safe bowl, heat granulated sugar and remaining ¼ cup (60 grams) water on high in 15-second intervals, stirring between each, until sugar dissolves. Let cool to room temperature.

Remove rugelach from oven, and immediately and generously brush with cooled syrup. (You should use most of the syrup.) Let cool on pans for 10 minutes. Remove from pans, and let cool completely on wire racks. Store in an airtight container for up to 5 days.