My understanding of biblical theology is that it is a theological understanding that is presented by a book of the Bible. Does it automatically become systematic theology when you pull from another book of the Bible if that other book is written by the same author?
Here's a definition of "biblical theology" from Wayne Grudem's popular Systematic Theology (pages 22-23):
"Biblical theology" has a technical meaning in theological studies. [...] [It] gives special attention to the teachings of individual authors and sections of Scripture, and to the place of each teaching in the historical development of Scripture. So one might ask, "What is the historical development of the teaching about prayer as it is seen throughout the history of the Old Testament and then of the New Testament?"
Here the difference between systematic and biblical theology is blurred, but he continues:
Yet there is still a difference, for biblical theology traces the historical development of a doctrine and the way in which one's place at some point in that historical development affects one's understanding and application of that particular doctrine. Biblical theology also focuses on the understanding of each doctrine that the biblical authors and their original hearers or readers possessed.
While systematic theology builds on biblical theology and may sometimes use its methods,
the focus of of systematic theology remains different: its focus is on the collection and then the summary of the teaching of all the biblical passages on a particular subject.
Thus, your original understanding of biblical theology is incorrect: it is not limited to a single book of the Bible. And by extension, no, biblical theology does not "automatically" become systematic theology just because multiple books of the bible are used. Systematic theology requires careful organization of biblical teaching on doctrines, with the specific goal of communicating those doctrines accurately, clearly, and concisely.