The best compost?

Yes, these are often grown in what looks like a mixture of clay and expanded rock bits. I think it is just the subsoil in the area where they are lifted from the fields in Italy.

You are right to query the type of compost to use as it would be better for the plant in a soil that matches. The nearest would be a loam type of compost such as John Innes No3. You could even make up your own or modify the above by adding a little extra grit and loam.