Step 1Place the oats in a blender or food processor and grind to a fine powder.
Step 2In the bowl of a stand mixer using a paddle attachment, or in a medium bowl using a hand mixer, mix together the butter, sugar, vanilla and salt until combined, 1 to 2 minutes, being careful not to overmix. Gently stir in the ground oats and flour and mix until thoroughly incorporated.
Step 3Divide the dough in half. Shape each half into an 8-inch log. Wrap each half tightly in plastic wrap. Chill at least 3 hours, up to overnight.
Step 4Heat the oven to 325 degrees. Slice each log crosswise into one-fourth inch slices. Place the slices on buttered baking sheets, spaced about 2 inches apart. Chill the baking sheets for 15 minutes, then place in the oven. Bake until the edges of the cookies just begin to brown, 16 to 18 minutes, rotating the sheets halfway through for even coloring. Cool the cookies on the sheets for 5 minutes, then move to a cooling rack to cool completely.