If you've had this trendy dish at a restaurant, you know "sabayon" refers to a simple frothy custard of egg yolks and wine (in this case, Champagne). The saltiness of oysters varies by species; taste one first, then season accordingly.
Place Champagne, 1/2 teaspoon sugar, and egg yolks in the top of a double boiler; stir with a whisk. Cook over simmering water until thick and mixture reaches 160° (about 5 minutes), stirring constantly with a whisk. Remove from heat. Stir in diced shallots, chopped thyme, kosher salt, and black pepper. Cool slightly.
Arrange oysters on a jelly-roll pan. Broil 3 minutes or just until edges begin to curl. Top evenly with sabayon. Garnish with thyme sprigs, if desired.