Mix together the butter, garlic, lemon juice, Italian seasoning, kosher salt, and pepper in a bowl.
Preheat the griddle to medium heat for a few minutes. Brush some of the butter on the griddle and add the asparagus. Cook the asparagus 3 to 4 minutes, turning a few times.
Add the shrimp with a little more butter, lightly season the shrimp and asparagus with kosher salt and pepper.
Cook another 3 to 4 minutes, turning a few times. Add the rest of the garlic butter, mix everything together, and cook another 1 to 2 minutes.