Apple-Oat Cookies

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2 1/4 cups all-purpose flour

1 cup old-fashioned oats

1 teaspoon ground cinnamon

3/4 teaspoon baking powder

1/2 teaspoon baking soda

1/2 teaspoon salt

1/4 teaspoon ground allspice

1 1/2 sticks unsalted butter, melted and cooled slightly

1/2 cup packed light brown sugar

1/2 cup granulated sugar

2 large eggs

1 Fuji or Gala apple, coarsely grated

2 teaspoons pure vanilla extract

1 cup confectioners' sugar

2 tablespoons milk