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Preheat the oven to 350°F. Lightly grease (or line with parchment) two baking sheets.
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Beat the butter, sugar, and vegetable oil in a bowl until combined, then add water and egg, beating until smooth. The mixture will look curdled at first, but will become creamy as you beat.
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Add the pudding mix, vanilla, salt, and baking powder, beating until smooth; the mixture will turn a lovely pastel green.
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Stir in the flour and nuts.
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Drop the dough by the tablespoonful — (a tablespoon cookie scoop works well here) —onto the prepared baking sheets, leaving about 2" between cookies.
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Flatten the cookies, if desired, using a fork, the flat bottom of a drinking glass, or the end of the pusher from your food processor. If you don't flatten the cookies, they'll bake up in softly flattened mounds; they're lovely either way.
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Bake the cookies for 15 minutes, until they're starting to brown on the bottom. Remove them from the oven, and cool right on the pans.
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Store, well -covered, for up to 5 days. Freeze for up to a month.