Combine water and 2/3 cup Amaretto, mix well and soak apricots in liquid for 6-8 hours (or overnight).
The next day…
Drain apricots and reserve liquid.
Heat butter in a 12-inch heavy nonstick skillet over medium heat until foam subsides, and then cook sugar, stirring constantly, until golden brown. Stir in reserved liquid (be careful; syrup will spatter) and simmer, stirring, 2 minutes.
Working in 2 batches, poach apricot halves in syrup at a low simmer, turning, 3 to 5 minutes per batch. Using a slotted spoon, transfer apricots, hollow sides up, to a platter.
Add additional 1/3 cup Amaretto liqueur, over low heat. (Be careful that the mixture does not ignite!)
Add crumbled biscotti to syrup and cook over low heat, crushing cookies with back of a wooden spoon, until melted into a coarse purée.
Gently simmer, stirring, until syrup is deep brown and slightly thickened. Cool syrup slightly.
Spoon syrup over apricots and sprinkle with toasted pine nuts. Serve warm or at room temperature.