I did not plant asparagus in a bed, but the old fashioned way in a trench. Asparagus roots grow upward so unless the roots are showing I would not fill in the bed until the crown has grown. The crown should not be buried. After the fronds come up you can add more compost around them. In the third year you should be able to get a small harvest.
I'm not worried about it coming up. It was beautiful last year. It's just that the soil has settled so much. I'm wondering if I should add more before the asparagus comes up, after the harvest, or just leave well enough alone.
I would wait until the asparagus fronds have come up and apply the compost then. I am not too concerned about the drop. Compost does that. It has to be added every year, I just like to be careful not to cover the crowns themselves.