Place the cherries in a heatproof bowl and cover with hot tap water. Let stand for 20 minutes to soften, then drain and transfer to paper towels to dry thoroughly.
Combine the almonds and sugar in a food processor and pulse until very finely ground. Add the almond extract, salt and egg whites, and pulse until the mixture comes together into a slightly sticky dough. Scrape the dough into a bowl, cover and refrigerate for at least 1 hour or up to overnight.
Heat the oven to 325 degrees. Line a baking sheet with parchment paper. Using a 1-ounce ice cream scoop or two tablespoons, portion a ball of dough. Using your finger, make an indentation into the ball and insert 1 cherry. Pinch the dough closed over the cherry, then roll to make a smooth, round ball. Repeat with the remaining dough and cherries.
Arrange 12 dough balls on the prepared baking sheet, spaced about 2 inches apart; cover the remaining dough balls until ready to bake. Slightly flatten each ball on the baking sheet with the palm of your hand. Bake the cookies, rotating the baking sheet halfway through, until lightly browned at the edges, about 20 minutes. Transfer to a rack and let cool completely before serving.
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