Step 1Heat the oven to 350 degrees. With a small, sharp knife or a melon baller, enlarge the cavity in each peach half where the pit was. Cut carefully, you don't want to break through the peach. Use about 1 teaspoon of butter to grease a small baking dish and arrange the peach halves in it, cut-side up. Chop the extra peach parts and put them in a mixing bowl.
Step 2Put the gingersnaps in a plastic bag and crush them lightly with a rolling pin. You don't want the crumbs to be too fine, they should be in small chunks. Add this to the chopped peaches. Add the egg yolks and sugar and stir vigorously with a wooden spoon to make a moist paste.
Step 3Divide the filling equally among the peach halves, mounding it in the center and spreading it gently to cover as much of the cut side of the peach as possible. Dot each peach with about 1 teaspoon of butter and sprinkle it with a little sugar.
Step 4Bake until the tops are brown and crisp and the peaches are soft and fragrant, 45 to 50 minutes. Serve warm, with a little lightly sweetened whipped cream or vanilla ice cream.