This recipe calls for homemade teriyaki sauce, which I think is better, but you can of course use a store-bought teriyaki sauce. The longer you marinate the doves in the sauce, the stronger tasting and saltier they will get. I like to keep them in the marinade only an hour.
Mix the sake, mirin, soy sauce, and sugar in a pot. Heat just to the point where the sugar dissolves. Turn off the heat and let cool. Submerge the birds in the marinade -- breast side down -- for up to an hour. If you go longer, the birds might get too salty.
Put the marinade in a small pot and boil it down until it gets syrupy. Alternatively, you can mix the corn starch with a little water and add that to the boiling sauce to thicken it. Either way, add lime juice to taste.
Get your grill nice and hot. Scrape down the grates and close the lid. Wet a paper towel with the vegetable oil and, using tongs, wipe down the grill grates. Paint the birds with sesame oil and put them breast side up on the grill, close the lid and cook over high heat for 4 minutes. Up this to 6 to 8 minutes for pigeon or squab.
Open the lid and paint the doves with the reduced marinade. Turn them on their sides and paint the exposed side. Grill over high heat (with the cover open) that way for another 1 to 2 minutes, or until the sides are nicely grilled. Repeat on the other side.
Finally, turn the doves breast side down on the grill and paint the tops. Grill for another 2 minutes (cover open), or until the skin is browned.
Remove from heat, paint one more time with the marinade, and let cool for 5 minutes. To serve, sprinkle the sesame seeds on top and have at it!