Chocolate Chip Ice Cream Sandwich

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4 sticks (1 pound) unsalted butter, at room temperature, plus more for greasing

4 1/2 cups all-purpose flour

2 teaspoons baking soda

2 cups softened butter

1 1/2 cups packed light brown sugar

1/2 cup granulated sugar

1 (3.4-ounce) package instant vanilla pudding mix

4 large eggs

3 cups semisweet chocolate chips

2 teaspoons pure vanilla extract

1 gallon creamy vanilla ice-cream (can substitute frozen yogurt), softened slightly

1 cup confectioners' sugar