Make the sorbet: Cook sugar and 1/2 cup water, stirring, in a small pan over medium heat until sugar has dissolved. Transfer to a bowl. Refrigerate until cold. Stir in lemon juice and Champagne. Freeze in an ice cream maker according to manufacturer's instructions. When mixture is half-frozen, whisk egg white until stiff (but not dry) peaks form; add to ice cream maker. Process until mixture is frozen. Transfer to an airtight container. Freeze at least 4 hours (up to 3 days).

Make the tapioca: Put large pearl tapioca, 3 cups milk, 3 tablespoons sugar, and 1 strip of lemon peel into a pan. Bring to a simmer over medium heat; cook, stirring occasionally, until tapioca pearls are tender and translucent, about 1 hour.

Meanwhile, put small pearl tapioca, the remaining 2 cups milk, 3 tablespoons sugar, and strip of lemon peel into another pan. Bring to a simmer over medium heat; cook, stirring occasionally, until tapioca is tender and pearls are translucent, about 30 minutes.

Put tapioca mixtures into a large bowl set over an ice-water bath; stir occasionally until cool, about 30 minutes. Cover with plastic wrap, pressing it directly onto surface. Refrigerate until cold, about 1 hour. To serve, stir in more milk to thin pudding if it has thickened too much. Divide pudding among serving dishes or glasses, and top with a scoop of sorbet.