In a medium saucepan, cover the beets with cold water. Add the thyme sprigs, black peppercorns and red wine vinegar and bring to a boil. Simmer, partially covered, until the beets are tender, about 45 minutes, replenishing the water if necessary. Drain the beets, then peel and cut them into 1/4-inch dice.
Return the diced beets to the saucepan. Add the sherry vinegar, sugar, rosemary sprig and 1/4 cup of water and bring to a boil. Cook over moderately high heat until a syrupy glaze forms, about 12 minutes. Discard the rosemary sprig and season the beets with salt.
Top each whole-grain toast with a spoonful of the glazed beets, a piece of burrata and a sprig of watercress. Drizzle with extra-virgin olive oil, garnish with the flaky salt and serve.