Is returning to speak with the first young man after vowing to cut ties with him considered breaking the vow? And what is the expiation for a vow to pray a specific number of rak'ahs at a specific time if it is not fulfilled on time?

It is impermissible for a woman to correspond with, befriend, or speak to non-mahram men due to the ensuing corruptions. If a vow is not fulfilled at its appointed time due to a legitimate Sharia-compliant excuse, such as illness, there is no sin, but it is obligatory to make it up, along with offering an expiation for a broken oath (kaffarat yameen) according to the Hanbalis. As for the Shafi'is and Malikis, they are only required to make it up. If the non-fulfillment was without an excuse, an expiation for a broken oath is obligatory according to the Hanbalis. Vows are disliked in Sharia. It is obligatory to repent and sever the unlawful relationship with both men, and to offer an expiation for a broken oath for the vow as a precautionary measure.