In cases of a non-Jew working on behalf of a Jew on Shabbat, there is a conceptual distinction – and sometimes a practical one – based on the type of work relationship that exists. A Jew may not have a worker who is paid on the basis of time (po’el) do melacha[1] on his behalf.[2] In contrast, it is fundamentally permitted to have a worker who is paid a set amount for the job (kablan) do melacha on Shabbat, as long as he is not required to do the work specifically on Shabbat.[3] However, the Rabbis did not allow even a non-Jewish kablan to do work on a Jew’s house (or in an equivalent public setting), as people might suspect that he is a Jew’s po’el (marit ayin).[4]
Although halacha is often based on broad categories and does not always change based on a case’s specific context, in this context, a reasonable amount depends on the extent to which people are practically likely to actually suspect him. For example, when the work is done outside the limits where Jews may go on Shabbat, it is permitted for a non-Jew to do the work as a kablan.[5]
Let us then consider your case practically. Even if people see your workers working and think you told them to do so, there should still be no problem. After all, when they see work being done a half hour before sunset on Friday afternoon, they will not suspect any wrongdoing, since as far as they know, you did not yet accept Shabbat.[6] This situation is reminiscent of the halacha regarding an Israeli spending a second day of Yom Tov that falls out on Friday in the Diaspora. Under ordinary circumstances, an Israeli who finds himself outside of Israel may not perform melacha on the second day of Yom Tov, even though he is only obligated to observe one day, due to the concern of marat ayin. However, the Radbaz[7] rules that he can cook on Friday without an eiruv tavshilin[8] because those who see him cooking do not know that he did not make an eiruv tavshilin.
Nevertheless, if this were the only point of leniency, we would be hesitant to rely on it. First of all, there is little discussion among the poskim regarding whether or not prohibitions of marit ayin apply during tosefet Shabbat.[9] Furthermore, considering you might return home from Shabbat davening with neighbors and make Kiddush before guests while hammers are still banging in the extension, there would seem to be a practical problem of marit ayin, as well.
However, there is a straightforward way to allow the work. The Rama[10] allows an individual to ask a non-Jew to do work for him after he has accepted Shabbat early. The Mishna Berura[11] confirms that this is permitted even when the need for the non-Jew’s help is not mitzva-related. (See the Shulchan Aruch,[12] who allows asking even another Jew to do work at this time if the latter did not yet accept Shabbat.) Accordingly, let us make a simple calculation. If it is permitted to ask a non-Jew to do work at that time, one can certainly allow a non-Jew to do work in his house even if it is seen publicly. After all, the reason for the general prohibition is that one might think that the homeowner hired the non-Jew as a po’el. The problem of a po’el is not that he is paid, but that he is doing working on a Jew’s behalf. Thus the worst thing that people might think you are doing – namely, that after accepting Shabbat early, you asked a non-Jew to do work for you – is in fact permitted. Thus, marit ayin obviously does not apply in this case.
That being said, if indeed the work is being done within the confines of your home and the members of your household see or hear the work, the work may not be in the proper spirit of Shabbat. It seems that the permission to employ a non-Jew during tosefet Shabbat is intended primarily for incidental, short-term work or work that is done away from the Jew’s proximity. Therefore, if it is possible, we would urge you to either begin Shabbat at the regular time during the period of construction or ask the workers to finish earlier than usual. However, if this causes significant problems, you may follow the purely halachic considerations that the restrictions upon what a non-Jew may do for you on Shabbat do not apply to tosefet Shabbat.