The custom during the Aseres Yemei Teshuvah (ten days from the first day of Rosh Hashanah through Yom Kippur) is to only eat Pas Yisrael. See here for a description and details of these laws by R. Doniel Neustadt. R. Neustadt writes:
The Shulchan Aruch(8) rules, however, that during aseres yemei teshuvah everyone should be careful to eat only pas Yisrael(9). There are several reasons,- all inter-related - for this halachah: a) so that we conduct ourselves with an extra measure of purity during these Days of Awe(10); b) to serve as a reminder of the unique status of these days(11); c) to beseech Hashem not to judge us stringently, just as we have adopted a practice which is not strictly required of us(12).Worthy of note is that while R. Neustadt quotes R. Yosef Shalom Elyashiv and R. Shlomo Zalman Auerbach as ruling that the laws of Pas Yisrael apply to cereal, the OU posekim disagree (link).
8 O.C. 603:1.
9 From the way the halachah is presented in Shulchan Aruch and Mishnah Berurah, it sounds as if it is a requirement. [See also Teshuvos Nachalas Shivah 72, who rules that it is an absolute obligation.] Chayei Adam 143:1, Kitzur Shulchan Aruch 130:2 and Aruch ha-Shulchan, though, quote this halachah as the "proper" thing to do, not as an obligation. See also Igros Moshe O.C.3:12..
10 In Talmudic times, everyone was careful not to allow their food to become impure (chullin b'taharah) during aseres yemei teshuvah; Tur quoting the Yerushalmi (Shabbos 3:3).
11 Levush O.C. 603.
12 Chayei Adam 143:1; Kitzur Shulchan Aruch 130:2. See also Elef ha-Magen 603:2, quoting the Ramak.
Some (including me) have the practice of also only eating and drinking Chalav Yisrael during this period. I wonder, however, whether this only makes sense according to those who explicitly consume non-Chalav Yisrael. Those who follow R. Moshe Feinstein, who was of the view that regular milk in the US is considered Chalav Yisrael, perhaps this stringency does not make sense. According to them, they always consume only Chalav Yisrael. Just thinking out loud...