Preheat oven to 350°F. Line a 9x13 baking dish with parchment paper, leaving overhang on two long sides. Grease exposed sides with butter or cooking spray.
Brown the butter: Place 1 cup butter in a small saucepan over medium heat, swirling occasionally. Butter will foam, then subside, and brown specks will form at bottom (about 5-7 minutes). Pour into a large mixing bowl immediately. Let cool 10 minutes.
Add sugar and 1/2 cup Biscoff spread to cooled brown butter. Whisk until smooth. Add eggs one at a time, whisking well after each. Whisk in vanilla. Mixture should be smooth and glossy.
In a separate bowl, whisk together flour, baking soda, and salt. Add half the dry ingredients to the wet mixture and gently fold with a spatula until just a few streaks of flour remain. Add half the buttermilk and fold. Repeat with remaining dry and buttermilk. Mix until batter is smooth and thick. Do not overmix.
Pour batter into prepared pan and spread evenly. Bake 25-30 minutes, until a toothpick inserted into center comes out with a few moist crumbs (not wet batter). Top should be golden and edges pulling away from parchment. The cake should spring back when pressed.
Place pan on a wire rack and let cake cool completely in the pan (about 2 hours). Do not frost a warm cake.
Make frosting: In a large bowl, beat softened 1/2 cup butter with 1 cup Biscoff spread until smooth. Add powdered sugar 1 cup at a time, beating on low. Add milk or cream and pinch of salt. Increase speed to medium and beat 2-3 minutes until light, fluffy, and spreadable. Add extra milk if needed.
Spread frosting evenly over cooled cake. Sprinkle crushed Biscoff cookies on top, pressing in gently. Let cake rest 15 minutes before slicing.