Immediately. Ten weeks accommodates the petition route with premium processing and leaves time to correct a problem; it is ample for the border route if the documents are already consistent. Resign and sign a lease only after the decision.
Ten weeks is enough for either door if the file is ready now
The calendar for a port-of-entry application is short: once the employer letter, degree and passport are in hand, the engineer can present himself at the border on a chosen date and be admitted the same day, so the real lead time is however long it takes to get a correct letter drafted and a credential evaluation if one is needed. The petition route requires the employer to prepare and file Form I-129; with premium processing, USCIS acts within its service window, but that action may be a request for evidence rather than an approval, and a response adds time. Working back from a start date ten weeks away, the petition should be filed within the first two weeks to leave room for a request, and the border visit should be scheduled after the approval notice arrives.
On either route, the engineer should not resign from the consulting firm, sign a U.S. lease or move the family until status is granted. A start date that the employer will not move is a reason to choose the route with the earliest certain decision, not a reason to skip preparation.