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Recent work by Huang, Lewis, Morales, Reiner, and Stanton suggests that the regular elliptic elements of $\mathrm{GL}_n \mathbb{F}_q$ are somehow analogous to the n-cycles of the symmetric group. In 1981, Stanley enumerated the factorizations of permutations into products of n-cycles. We study the analogous problem of enumerating the factorizations of various elements of $\mathrm{GL}_n \mathbb{F}_q$ into products of regular elliptic elements. More precisely, we define a notion of cycle type for $\mathrm{GL}_n \mathbb{F}_q$ and seek to enumerate the tuples of a fixed number of regular elliptic elements whose product has a given cycle type. In some special cases, we provide explicit formulas, using a standard character-theoretic technique due to Frobenius. We also address, for large q, the problem of computing the probability that a random tuple of regular elliptic elements has products with a specified cycle type. We conclude with some results about the polynomiality of our enumerative formulas and some open problems.