Prepare a medium charcoal or gas grill fire. Brush the grill grate clean.
In a small bowl, whisk the butter, jam, Chambord, sugar, and salt until well combined. Halve and pit the peaches. If the pit leaves a very shallow well, scoop out a little more flesh with a melon baller or a measuring spoon. Brush the cut sides of the peaches lightly with the oil.
Grill the peaches cut side down (covered on a gas grill) until nicely grill-marked, 3 to 4 minutes. Flip the peaches and spoon the raspberry butter into the pit well of each peach half (you may or may not need to use all the butter). Cover and grill until the butter is melted and the peaches are soft and cooked through, 2 to 3 minutes more. Serve warm or at room temperature with a scoop of vanilla ice cream topped with the biscotti or amaretti crumbles.