Preheat oven to 375°F (190°C). Grease or line a 12-cup muffin tin with paper liners. In large bowl, whisk together flour, sugar, baking powder, baking soda, salt and spice. In a separate bowl, whisk together egg, buttermilk and oil. Pour wet ingredients and sesame seeds into flour mixture; stir until just combined.
Evenly spoon half the batter into the muffin tins. Add 1 tsp (5 mL) apricot jam into each tin, then top with remaining batter.
Bake on middle rack of oven until tops are firm to the touch, about 20 min. Cool in tin 5 min. then transfer muffins to rack to cool completely.
While muffins are cooling, in small bowl, mix together icing sugar and milk until smooth. Drizzle on muffins. Allow a few minutes for icing to set. Store muffins in airtight container up to 2 days or wrap individually and freeze up to 2 weeks.