Lightly butter a 10- by 15-inch rimmed baking pan. Line pan with a 15- by 20-inch piece of cooking parchment, spreading it smooth (edges will stick up above pan rim). Butter and flour parchment.
In a 2- to 3-quart pan over medium-high heat, melt 1/3 cup butter, stirring often, about 2 minutes. Remove from heat and add sugar, corn syrup, vanilla, and salt; stir until smooth. Add oats and coconut; mix until evenly blended. Scatter oat mixture in lined pan; with a long metal spatula, spread mixture level.
Bake in a 375° regular or convection oven until cookie is browned at edges and bubbling all over, about 10 minutes. Let cool just until cookie begins to firm but is still warm, about 2 minutes (if too cool, cookies will crack when cut). Invert a 12- by 15-inch baking sheet over pan; holding pan and sheet together, carefully invert cookie onto sheet. Cut into 18 rectangles 2 1/2 by 3 1/3 inches. Let cool.
Return 9 of the cookies, smooth side up, to lined pan, setting them side by side in a single layer to make a snug rectangle in half the pan.
With scissors, snip through opposite sides of ice cream carton from top to bottom; peel carton from ice cream and discard. Working quickly, place ice cream on a board (if round, trim off 1 side and lay on flat side). With a long, heavy knife, cut crosswise into 1/2-inch-thick slices. Set slices side by side over cookies in pan; fill in any bare spots with ice cream scraps to cover completely, and trim off any excess. Freeze remaining ice cream for other uses. Cover ice cream with remaining 9 cookies, matching the orientation of bottom cookies. Wrap pan airtight and freeze until ice cream is firm, at least 3 hours.
Lift ice cream sandwiches on parchment from pan and transfer to a board. Cut between cookies; trim sandwich edges with a knife, if desired. Serve, or wrap sandwiches individually airtight and freeze up to 2 weeks.
Nutritioal analysis per sandwich.