For the cookies: Position racks in the upper third of the oven. Preheat the oven to 350°F. Line 2 jelly-roll pans with parchment.

1) In a small saucepan, melt the butter over medium heat. Once melted, continue to simmer over medium-low heat until the liquid turns a deep, rich golden color, 3 to 5 minutes. The milk solids, which sink to the bottom, will turn very dark brown; do not overcook and allow the mixture to burn. Skim away any foam on the surface.

2) Pour the browned butter into a large bowl, leaving the milk solids behind. Whisk in the brown sugar and sugar until combined. Allow to cool briefly, then whisk in the egg and vanilla. In a separate bowl, whisk together the oats, flour, baking soda, cinnamon, and salt. Add the dry mixture to the wet ingredients and stir until just combined.

3) Place scant 2-tablespoon-size scoops of cookie batter on the pans, placing them 2 inches apart (just make sure you have 16 equal-size cookies). Do not flatten. Bake for 10 to 14 minutes, or until the cookies are a light golden brown around the edges, but still a little soft, but not wet, in the center.

4) Cool on the pans set on a rack for 5 minutes, then remove the cookies to a cooling rack and cool completely. Store at room temperature for up to 1 day in an airtight container.

For the filling: Spread the blueberries in a single layer in a nonstick skillet. Sprinkle with the sugar and water. Stir to coat. Heat over medium heat until the sugar begins to dissolve, then mash with a potato masher. Bring to a simmer and continue to cook, stirring occasionally, until the juices have begun to reduce and thicken, about 4 minutes. The mixture should be a deep, dark purple, with a syrupy consistency. It should be thick enough so that when you draw a spoon along the bottom you can see the pan for a few moments. Remove from the heat and cool completely.

1) Remove the ice cream from the freezer and allow to sit at room temperature for 5 to 10 minutes, or until softened but not melted. Fold in the cooled blueberry mixture until it is a uniform color. If you have the time, it is best to chill the ice cream at this point until it is solid again. The blueberry ice cream can be made a day ahead and frozen overnight in an airtight container.

For the assembly: Have a tray in the freezer that will fit all 8 of the ice-cream sandwiches. Remove the ice cream from the freezer and let soften at room temperature for a few minutes. Place a generous 2 tablespoon-size scoop of ice cream on a bottom side of a cookie and top with another cookie, bottom side against the ice cream. Squeeze gently so that the ice cream presses out to fill the entire cookie surface. Immediately place on the tray in the freezer. Repeat with the remaining cookies and freeze until solid. Place the sandwiches in a zipper-top plastic bag and place back in the freezer until needed. Remove the sandwiches from the freezer about 5 minutes before serving. The sandwiches may be frozen for up to 3 days in zipper-top bag or an airtight container.