For dough: In a small saucepan, heat ¾ cup whole milk and ½ cup water over medium heat until bubbles begin to form around sides of pan and an instant-read thermometer registers 180°F (82°C). Remove from heat, and add ¾ cup unsalted butter. Let cool, stirring occasionally, until the butter is melted and an instant-read thermometer registers 120°F (49°C) to 130°F (54°C).
In the bowl of a stand mixer fitted with the paddle attachment, combine 2 cups all-purpose flour, ¼ cup granulated sugar, 1 tablespoon kosher salt, and 1 package active dry yeast. Add warm milk mixture, and beat at low speed until combined. Add 1 large egg, beating until combined.
With mixer on low speed, gradually add 2 cups all-purpose flour, beating just until a shaggy dough comes together and stopping to scrape sides of bowl.
Switch to the dough hook attachment. Beat at low speed until a soft, slightly sticky dough forms, about 24 minutes, stopping to scrape sides of bowl and dough hook; add up to remaining ¼ cup all-purpose flour, 1 tablespoon at a time, if dough is too sticky. Turn out dough onto a clean surface; knead 5 to 6 times, 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, 1 to 2 hours.
Preheat oven to 375°F (190°C).
For strawberries: On a small rimmed baking sheet, stir together 1½ pounds hulled fresh strawberries, 2 tablespoons granulated sugar, and ½ teaspoon kosher salt until well combined; spread in a single layer. Bake until berries are very soft and fragrant and have released ample juices, about 30 minutes, stirring every 10 minutes.
Strain strawberries and juices through a fine-mesh sieve set over a small saucepan. Let strawberries cool for 15 to 20 minutes. Cut into ¼-inch pieces. Transfer any additional berry juices to saucepan.
In a medium bowl, stir together chopped strawberries, ¼ cup melted butter, and 2 teaspoons vanilla bean paste.
In a small bowl, place ½ cup plus 2 tablespoons granulated sugar. Add 3 tablespoons water, remaining ¼ cup melted butter, and remaining ¼ cup granulated sugar to juice in saucepan.
Lightly punch down dough. Cover and let stand for 10 minutes. Spray a 3-quart baking dish with baking spray with flour.
Turn out dough onto a lightly floured surface, and divide in half (about 495 grams each). Roll each portion into a 30-inch rope. Cut each rope into 1-inch-long pieces (about 16 grams each). Working in batches, dip dough pieces in strawberry mixture; dredge in granulated sugar, gently shaking off excess. Place dough pieces in pan in even layers. Pour remaining strawberry mixture all over dough. 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).
Bring juice mixture in saucepan to a boil over medium-high heat, stirring occasionally until butter is melted and sugar dissolves; cook for 2 to 3 minutes. Pour all over dough.
Bake until golden brown and an instant-read thermometer inserted in center registers 190°F (88°C), 30 to 35 minutes, loosely covering with foil to prevent excess browning, if necessary. Let cool on a wire rack for 20 minutes.
For glaze: In another medium bowl, whisk together ½ cup confectioners’ sugar, 1 tablespoon whole milk, 1 tablespoon melted butter, and 1 teaspoon vanilla bean paste until smooth. Drizzle all over warm bread. Serve immediately. Cover and refrigerate for up to 3 days.
