In a large bowl, whisk together the egg, egg yolks, brown sugar, and granulated sugar until smooth and well combined. Set aside.
In a large saucepan, combine the heavy cream, half-and-half, and kosher salt. Heat over medium heat, stirring occasionally, until the mixture is steaming and just beginning to simmer around the edges. Do not let it come to a full boil.
Slowly ladle about 1 cup of the hot cream mixture into the egg mixture, whisking constantly to temper the eggs. Continue adding another cup of the hot cream, whisking continuously.
Pour the tempered egg mixture back into the saucepan with the remaining cream. Cook over medium-low heat, stirring constantly with a wooden spoon or heatproof spatula, until the custard thickens slightly and reaches 170°F on an instant-read thermometer. It should coat the back of a spoon, and when you run your finger through it, the line should remain clean.
Remove the pan from the heat and immediately whisk in the vanilla extract, vanilla bean paste, and warmed cookie butter until completely smooth and fully incorporated.
Pour the custard through a fine-mesh sieve into a clean bowl to remove any cooked egg bits. Press a piece of plastic wrap directly onto the surface of the custard to prevent a skin from forming.
Refrigerate until completely chilled, at least 6 hours but preferably overnight.
Pour the chilled custard into your ice cream maker and churn according to the manufacturer's instructions until it reaches the consistency of thick soft serve.
To assemble, transfer half of the ice cream to a freezer safe container. Then swirl in half of the warmed cookie butter and sprinkle with half of the crushed Biscoff cookies. Repeat with the remaining half of the ice cream. Drizzle the remaining warmed cookie butter and swirl on the top then sprinkle with crushed cookies.
Seal and freeze for 6-8 hours or overnight, or until firm enough to scoop.
Serve topped with crushed Biscoff cookies and an extra drizzle of warmed cookie butter, if desired.