Use a small screwdriver to pierce coconut's "eyes." Drain liquid; discard. Place coconut on a folded kitchen towel; turn and gently tap with a hammer a few inches below eyes to split. Break coconut into pieces, remove white meat, and cut away brown skin. Thinly slice coconut meat on a mandoline or with a vegetable peeler. Measure 1 cup.
Preheat oven to 350 degrees, with rack in lower third. Butter sides of twelve 4-ounce canning jars or ramekins, and coat with granulated sugar; tap out excess. Place jars on a baking sheet.
In a small saucepan, melt butter over medium-low heat. Add flour, and whisk constantly for 3 minutes. Gradually whisk in coconut milk. Bring to a boil, whisking until thickened, 2 to 3 minutes. Remove from heat, and whisk in 3 tablespoons confectioners' sugar, the vanilla and coconut extracts, and lime juice. Transfer to a medium bowl.
In the bowl of a mixer fitted with a whisk, beat egg whites and cream of tartar on medium until soft peaks form, about 3 minutes. Gradually add remaining 3 tablespoons confectioners' sugar. Raise speed to high; beat until stiff peaks form, about 1 minute. Whisk reserved coconut mixture to loosen; whisk one-third of egg whites into it to lighten. Gently fold in remaining egg whites. Pour into prepared jars or ramekins, filling to 1/4 inch below rims. Place several strips of coconut on each; add stickers to jars, if using. Bake until puffed and golden brown, about 15 minutes. Dust with confectioners' sugar, and serve immediately.