Preheat only one side of the barbecue to maximum heat, or to a temperature of 200°C (400°F).
Place a roasting pan on the unheated side of the grill.
With the tip of a knife, slice diamond-shaped incisions into the skin of the duck breasts, without cutting into the meat.
Season with salt and pepper. Set aside.
In a saucepan, mix together all the remaining ingredients except the cornstarch.
Bring to a boil over medium-high heat and reduce by half.
Dissolve the cornstarch in a bit of lukewarm water and pour into the sauce while stirring. Simmer for 1 minute and set aside.
Place the breasts in the roasting pan, skin side up. Close the barbecue lid and cook for about 6 minutes.
Brush with the glaze.
Turn the breasts over and brush the meat side. Cover and cook for about 6 minutes.
Repeat the previous steps and cook for 3 more minutes on each side.
To complete the cooking and make the breasts a bit crispy, transfer the duck onto the heated side. Brush again and cook for about 2 minutes on each side.
Remove the breasts and cover with aluminum foil.
Let rest for 5 minutes.
Slice the breasts and place them on plates.
Serve immediately with hash browns fried in duck fat and grilled asparagus.