Drawing is Reagan's favorite thing to do. The gerund phrase is the direct object of the verb is.
A gerund phrase starts with a gerund, a word with ING, and includes other modifiers or objects. Gerund phrases always function as nouns. Thus, gerund phrases can either be subjects, subject complements, or objects in the sentence.
To further prove that the gerund phrase serves a the direct object of the verb is, we can rewrite the gerund phrase into a proper sentence.
From: Drawing is Reagan's favorite thing to do.
To: Reagan's favorite thing to do is drawing.
Subject: Reagan's favorite thing to do
Verb: is
Object: drawing