Line a 9x13-inch pan with parchment. Lightly grease the paper with the oil.
Melt the butter in a large pot over low heat.
Stir in the cookie butter and marshmallows. Cook gently, stirring, until smooth and glossy.
Remove from heat. Mix in the vanilla and salt.
Fold in the cereal and crushed cookies until evenly coated. Work quickly.
Press the mixture into the pan with a buttered spatula. Do not pack too firmly.
Drizzle with melted white chocolate and warmed cookie butter. Swirl, cool until set, then cut into squares.