FWIW, I would not leave job 1 for job 2 without, at least, a written offer letter from job 2 which included a firm start date. Of course, if you are to be hired as an "at-will" employee, then job 2 could withdraw the offer notwithstanding the offer letter. Nonetheless, if a company issues an offer letter, it is usually committed to having the prospective employee start working on the start date included in said letter. In contrast, a verbal offer to start in a week or two appears rather casual at best or uncomfortably shaky at worst. Why leave the job you actually have on the basis of an amorphous promise for a job on an unspecified date?
Of course, you can "ask" for an employment contract. However, if your prospective employer planned to hire you on an at-will basis, virtually no chance exists it would then voluntarily consent to providing you with a written employment agreement. Indeed, as stated above, your prospective employer's representatives may consequently reconsider whether you are the "right fit" for them at all.
