Bring strawberries, sugar, lemon juice and peppercorns to a boil. Let simmer for 10-15 minutes until slightly thickened. Strain through a mesh sieve and let cool completely.

While the berries simmer, prepare the chocolate ice cubes: Bring all the ingredients to a low simmer in another pot, stirring constantly. When the chocolate is completely melted, pour into ice cube trays and freeze.

When ice cubes are frozen and syrup is chilled, place a few ice cubes in the bottom of a glass. Mix 2 cups of chilled milk and 2 tablespoons of syrup (or more, according to taste). Pour strawberry milk over the ice cubes.