It is indeed capturing a Hebraism. In Hebrew they would use the infinitive plus the relevant verb to express to a greater degree the action or emotion.
In Genesis 2:17, for the earliest example I'm aware of in Scripture, the Hebrew is מות תמות ("dying thou shalt die;" lit. "to die thou shalt die"). This is translated in Greek as θανάτῳ ἀποθανεῗσθε in the Greek Old Testament (dative, indicating "by" or "with" or "to," followed by the relevant tense verb, here future) which is best translated "thou shalt die the death."
The New Testament authors are no stranger to Hebraisms—the New Testament is almost written in a Greek unto itself, quite distinct from normal Koine Greek (inasmuch as Hebrew concepts and usages of words are imported to the point of making it quite difficult to understand for someone ignorant of Hebrew culture—it's probably best called Jewish Greek: a non-Jew would perhaps not know what's going on here with "with desire I have desired," and might consider it butchered Greek!).
For example, prayers or songs or sentences uttered in Hebrew, in order to indicate that they were spoken in said language, are usually recorded with the omission of the usual grammatical functors (most often the verb "to be").
E.g. "Hail, favored one! the Lord is with thee; blessed art thou among women" in Luke 1:28 in Greek lacks any "to be" verbs, and uses simply "μετα" ("with") to mimic the fact that in Hebrew, you would also omit the "to be" verb when saying "the Lord is with thee" and simply say "The Lord immak" ("the Lord with-you").
Therefore, if Christ spoke Hebrew, he would have said, "נכסף נכסףתי" (nikhsof nikhsafti) (desiring I have desired) which means, "I have greatly desired." Or in modern English, "I've been waiting so long to eat this Passover with you!"
The aorist is simply used because it denotes in Greek what Jesus did "with desire."