In a medium bowl, whisk the flour and salt. In a large bowl, using an electric mixer at medium speed, beat 3/4 cup sugar and the butter, occasionally scraping down the bowl, until light and fluffy, about 1 minute. Add the egg and vanilla; beat to blend. Reduce the speed to low; add the flour mixture and beat until just blended. Cover and refrigerate until firm, about 1 hour.
In a small saucepan, cook the pineapple and the remaining 1/4 cup sugar over medium, stirring often, until the pineapple is translucent and the mixture is thick, about 12 minutes. Transfer to a wide bowl. Stir in the cherries and let cool.
Position the racks in the upper and lower thirds of the oven; preheat to 350 degrees . Roll dough by the tablespoon into balls. Place 1 inch apart on 2 parchment-lined baking sheets. With your thumb, make a deep indentation in the center of each ball. Fill the indentations with the pineapple mixture.
Bake the cookies until golden around the edges, 20 to 25 minutes. Transfer the cookies, on the parchment, to racks and dust with confectioners' sugar; let cool completely.