Warm the milk, sugar, cookie butter, 1 cup of the heavy cream and the salt in a medium saucepan over medium-low heat, stirring until the sugar is dissolved and the cookie butter has melted. Continue cooking until the mixture comes to a slight simmer, bubbling around the edges of the pan.
Meanwhile, pour the remaining 1 cup heavy cream into a large bowl and place a fine-mesh sieve on top. Whisk the egg yolks in a medium bowl. Slowly pour the warmed milk mixture into the egg yolks, whisking constantly. Scrape the mixture back into the saucepan.
Place the saucepan over medium heat and stir constantly with a rubber spatula, being sure to scrape the bottom of the pan as you stir, until the mixture thickens and coats the back of the spatula, a few minutes. The mixture should register 170 to 175 degrees F on an instant-read digital thermometer.
Pour the custard through the fine-mesh sieve and stir it into the cream. Stir in the vanilla extract and place the bowl over an ice bath. Stir occasionally, until the mixture is cool. Cover and transfer the custard to the refrigerator until completely chilled, at least 8 hours or overnight.
Freeze the mixture in your ice cream maker according to the manufacturer's instructions, adding the chopped speculoos cookies during the last minute of churning. As you remove the ice cream to a freezer-safe container, drop dollops of the melted cookie butter over each layer of ice cream. Once all of the ice cream and cookie butter has been placed in the container, use a butter knife to gently swirl the mixture. Store the ice cream in the freezer.