Melt 3 tablespoons butter in a saucepan over medium-high heat. Add chicken thighs with the smoother sides facing down and cook until browned, 3 to 4 minutes. Flip and repeat on the other sides. Turn off the heat and remove chicken to a plate. Let sit until cool enough to handle, about 10 minutes.

Cut chicken into 1/2-inch pieces and transfer back to the plate.

Pour chicken and any accumulated juices back into the saucepan. Season with salt, pepper, and cayenne. Stir in hot sauce, turn heat to medium-high, and bring to a simmer. Reduce heat to medium and simmer, stirring occasionally, until slightly reduced, 10 to 15 minutes. Transfer to a bowl to cool.

Meanwhile, combine eggs, salt, flour, and milk for Dutch babies in a bowl. Whisk for a few minutes until batter is smooth and thin. Let rest for 15 to 20 minutes.

Grease a 12-cup muffin tin with 1 tablespoon butter, then spray the cups with nonstick spray.

Transfer the rested batter into a liquid measure. Pour into the prepared muffin cups, filling each about halfway full. Spoon some Buffalo chicken into each cup then sprinkle blue cheese over top.

Place in the center of a cold oven. Set a baking sheet on a lower rack to catch any drips. Turn the heat to 450 degrees F (230 degrees C) and set a timer for 30 minutes, at which point the Dutch babies should be puffed up and browned.