Profiteroles With Maple Caramel Sauce

¾ cup (1½ sticks) unsalted butter
¾ cup whole milk
1 Tbsp. granulated sugar (optional)
1 tsp. Diamond Crystal or ½ tsp. Morton kosher salt
1½ cups (188 g) all-purpose flour
5 large eggs
1 large egg, separated
Sauce and assembly
4 Tbsp. unsalted butter
¾ cup heavy cream
¾ cup pure maple syrup
1¼ tsp. Diamond Crystal or ¾ tsp. Morton kosher salt
6 oz. white chocolate, coarsely chopped
Vanilla bean ice cream (for serving)

Place racks in upper and lower thirds of oven; preheat to 425°. Heat ¾ cup (1½ sticks) unsalted butter, ¾ cup whole milk, 1 Tbsp. granulated sugar (optional), 1 tsp. Diamond Crystal or ½ tsp. Morton kosher salt, and ¾ cup water in a medium saucepan over medium-high, whisking occasionally to prevent scorching, until butter is melted and mixture is boiling (it should bubble vigorously, but watch it closely so it doesn’t boil over). Immediately remove pan from heat and add 1½ cups (188 g) all-purpose flour all at once. Whisk until incorporated and a paste forms. Return to medium-high heat and cook, mixing vigorously with a wooden spoon, until dough is the consistency of smooth mashed potatoes, about 30 seconds. Continue cooking, stirring constantly, until dough starts to dry out, creates a film on bottom of pot, and releases a lot of steam, about 3 minutes.
Transfer dough to the bowl of a stand mixer fitted with the paddle attachment and beat on medium speed 1 minute. Reduce speed to medium-low and add 5 large eggs one at a time, beating to combine after each addition. Scrape down sides of bowl with a rubber spatula. Beat in 1 large egg white. Increase speed to medium and beat until dough holds medium-soft peaks when you lift paddle out of dough, 8–10 minutes. (The dough should be at or just above room temperature and texture should be visibly voluptuous, glossy and very smooth. When you pinch a bit between your thumb and index finger, you should be able to bring your fingers apart and have the dough stay connected by a thread.)
Beat 1 large egg yolk with 1 tsp. water in a small bowl to blend; set egg wash aside.
Scrape dough into a pastry bag fitted with a round ½" tip or a resealable plastic bag with 1 corner cut on a diagonal to make a ½" opening. Pipe 1½"-diameter rounds between 2 parchment-lined baking sheet, spacing about 1" apart. (Piping rows of 4 down the short sides and 5 across long sides works well.) Brush tops with reserved egg wash.
Bake profiteroles 10 minutes, then reduce oven temperature to 350° (do not open oven) and bake 15 minutes. Rotate baking sheets top to bottom and front to back, then continue to bake until golden brown and firm to the touch, 5–7 minutes more. Let cool.
Sauce and assembly
While the profiteroles are baking, melt 4 Tbsp. unsalted butter in a small saucepan over medium heat. Add ¾ cup heavy cream, ¾ cup pure maple syrup, and 1¼ tsp. Diamond Crystal or ¾ tsp. Morton kosher salt and bring to a rolling boil. Cook, whisking often and adjusting heat as needed so the mixture doesn’t boil over, until thickened to a the consistency of pancake syrup and a cloudy amber in color, 5–8 minutes. Remove from heat. Add 6 oz. white chocolate, coarsely chopped, and whisk until chocolate is melted and mixture is smooth. Cover pan and keep sauce warm until ready to use.
To serve, cut profiteroles in half horizontally and scoop a rounded tablespoonful vanilla bean ice cream into each bottom half. Cover with top halves. Divide profiteroles among plates as desired and spoon maple sauce over.