Preheat the oven to 275 degrees. Line two baking sheets with parchment paper.
In a small, heavy-bottomed saucepan, add the apple, bourbon, dark brown sugar and vinegar. Cook over medium-high heat until the mixture boils and liquid reduces, about 3 minutes. Reduce the temperature to low and cover. Stir the apple mixture every few minutes, until the apples have darkened and caramelized, 8 to 10 minutes. Remove from the heat and let cool.
In a food processor, pulse the walnuts and powdered sugar until the walnuts are finely ground, about a minute. Add the thyme and pulse for 10 seconds.
In a standing mixer or using a hand-held mixer, beat the egg whites, cream of tartar and salt until white and almost stiff, then slowly add the cane sugar and then the vanilla. Increase the speed until the sugar is dissolved and the egg whites are thick and glossy. Remove the beaters and gently fold in the walnut mixture in 3 additions.
Fit a piping bag with just the base tip, so that the opening is ¾-inch wide. Rest the bag tip down in a cup while filling it. Use a rubber spatula to scoop the cookie mixture from bowl to bag, then twist the top to keep the meringue from spilling out. Holding the bag upright, squeeze gently to pipe a 1-inch circle. Lightly lift the bag to create a small peak. After all the rounds have been piped, place one small apple piece over each peak.
Bake for 25 minutes, then turn off the heat and let the meringues cool in the oven for an hour.
To maintain the crisp texture, leave the meringues loosely covered (not airtight) on a plate or in a bag. Makes approximately 3 dozen.