For the oats:

Melt the butter in a small nonstick saucepan over medium heat until it stops foaming. Add the oats and cook, stirring constantly, until medium golden brown and fragrant, about 6 minutes. Standing back because the mixture will spit and sputter, add 2⁄3 cup milk, the water, and salt and bring to a boil. Reduce the heat to medium-low and simmer, stirring occasionally, until the oats are very soft and almost all of the liquid has been absorbed, about 18 minutes. Off the heat, cool the mixture to room temperature, about 20 minutes.

For the shake:

Place 2/3 cups (5 ounces/142 grams) of the cooked oats, the remaining 1/2 cups (4 ounces/125 milliliters) milk, the brown sugar, and cloves in a blender and blend until the oats are completely broken down and incorporated into the milk, about 1 minutes. Add the ice cream and pulse several times to begin breaking it up. With the blender motor off, use a flexible spatula to mash the mixture down onto the blender blades. Continue pulsing, stopping, and mashing until the mixture is well blended, thick, and moves easily in the blender jar, roughly 30 to 90 seconds. Pour into a chilled glass or glasses, and serve at once.

To make this shake with Scotch and honey, substitute 2 tablespoons (about 1 ounce/30 milliliters) Scotch whiskey for 2 tablespoons of the 1/2 cup milk in the shake, reduce the brown sugar to 2 tablespoons (about 1 ounce/36 grams), and add 2 tablespoons of honey (about 1 1/2 ounces/43 grams) to the blender along with the ice cream.