According to UML syntactical rules, both solutions are valid - both class attributes and associated classes are so called class properties and can be shown as attributes (inside the class) or as separate classes, linked via association. For both these class features you can define name, multiplicity, scope, etc. Please refer to UML spec for detailed technical information.
However, the common practice if the following:
- if the property is basic data type (int, boolean, date, etc) -> show
it inside a class, as attribute
- if the property is full fledged
class -> show it as a separate class entity and use association to
display their relationship
This practice make sense, as "int", "boolean" or "date" do not have their own custom properties and it is enough to show them inside a class (withoult losing information about them). Classes, on the other side, have their own features (attributes, methods and own associations, generalizations, etc) and therefore "deserve" more space on the diagram.
Concluding with the direct answer to your question: show Doctor as a separate class on the diagram, connected with Patient via association (note the property name displayed as associationEnd name). Keep both dates inside the Patient class:
The following diagram is equivalent and valid, but you might agree that the first one is visually clearer (imagine some atts, methods and relationships or the Doctor class) and therefore recommended:
UPDATE (after the comments)
Note: Composition is used for Dates, to reflect a strong relationship of the Whole-Part kind and the fact that these Dates cannot be unlinked from their context.
The other association (Patient-Doctor) is a common assotiation and the corresponding link can be broken anytime (for example to change a Patient's Doctor).