Place milk, whipping cream, vanilla, salt, and 3/4 cup of the sugar in a medium saucepan; bring to a simmer over medium. Simmer, whisking constantly, until sugar is dissolved, about 3 minutes. Transfer mixture to a medium-size heatproof bowl. Place a piece of plastic wrap directly on surface of mixture. Chill at least 2 hours or up to 24 hours.

Meanwhile, preheat oven to 275°F. Spread cake pieces in an even layer on a small rimmed baking sheet. Bake in preheated oven until lightly toasted, about 30 minutes. Let cool 15 minutes; crumble. Set aside until ready to use or up to 24 hours (wrap tightly with plastic wrap if storing overnight).

Place diced strawberries and remaining 1/2 cup sugar in a medium saucepan; bring to a boil over medium. Boil, stirring occasionally, until strawberries begin to break down and sauce slightly thickens, about 10 minutes. Cover and refrigerate at least 30 minutes or up to 24 hours.

Pour chilled milk mixture into frozen freezer bowl of a 2 1/2- to 3-quart electric ice-cream maker. Freeze according to manufacturer’s instructions. (Instructions and times may vary.)

Spoon half of the ice cream into a freezer-safe container; spoon half of the chilled strawberry sauce over ice cream, and sprinkle with half of the cake crumbles. Repeat layers once using remaining ice cream, strawberry sauce, and cake crumbles. Cover and freeze until firm, about 2 hours. Serve with halved fresh strawberries.