Line a 9-inch springform pan with parchment paper or set out a deep dish pie plate.
Crush 3 cups of the Cheerios into uniform, fine crumbs in a food processor or with a rolling pin in a zipper top bag. Toss the crumbs and butter together in the food processor or in your prepared pan. Press evenly over the bottom of the pan. Set aside.
In a stand mixer fitted with a whisk attachment or a mixing bowl with a hand mixer, whip the cream cheese and vanilla extract together on HIGH until light and fluffy. Add in the squishy chocolate ice cream and whip until even, scraping down the sides of the bowl as needed. Transfer the filling to the prepared crust and spread evenly. Place in the freezer for at least 4 hours, or until frozen solid. After 4 hours, place a piece of plastic wrap directly on the surface if you do not plan on eating it that day.
When you are ready to serve, let the icebox pie stand at room temperature for about 10 minutes before unmolding it if you used the springform pan. Pile up the remaining Chocolate Cheerios in the center of the icebox pie.
Pipe or spread the whipped cream or whipped topping around the Cheerios, drizzle lightly with chocolate syrup, and grate some chocolate on a zester or fine cheese grater over the whole icebox pie. Slice and serve immediately.
If you have leftovers, place back in the freezer uncovered until the whipped cream or whipped topping is firm, then cover with plastic wrap.