The Men of the Challah are right!

As a satirical response to the Women of the Wall, they write:

As Men of the Challah, our central mission is to achieve the social and legal recognition of our right, as men, to make challah, light candles, and not get up early for Shacharis minyanim.

Cute. But there is no reason that men should not bake challah for Shabbos. Here is Aruch Hashulchan, Orach Chaim, 250, seif 3:

וכתב הרמב"ם בפרק ל' דין ו':

"אף על פי שהיה אדם חשוב ביותר ואין דרכו ליקח דברים מן השוק ולא להתעסק במלאכות שבבית - חייב לעשות דברים שהן לצורך השבת בגופו, שזהו כבודו. וחכמים הראשונים מהם מי שהיה מפצל העצים לבשל בהן, ומהם מי שהיה מבשל או מולח בשר או גודל פתילות או מדליק נרות, ומהם מי שהיה יוצא וקונה דברים שהן לצורך השבת ממאכל ומשקה, אף על פי שאין דרכו בכך. וכל המרבה בדבר זה - הרי זה משובח" עכ"ל. (שבת קי"ט.) ולא יאמר 'כי אפגום כבודי' - שזה הוא כבודו, שמכבד את השבת.

וכתב בזה לשון חיוב, והטור והש"ע לא הזכירו לשון חיוב ע"ש. ואף הוא אין כוונתו לחיוב גמור, שהרי רבים וגדולים אין עושין בעצמן. ועוד דבש"ס שם חשיב מי שהיו עושין בעצמן - שמע מינה שרבים לא עשו בעצמן אלא הוא הידור מצוה. ואצלינו נשותינו עושות ומכינות בעצמן לכבוד שבת, ואשתו כגופו. (עיין קונטרס אחרון להגר"ז וצ"ע ודו"ק)

"And the Rambam wrote in perek 30 din 5:

Even though a person (man) is exceptionally distinguished such that it is not his way to purchase items from the market nor to engage in the housework, he is obligated to perform things which are for the sake of Shabbos with his own body, for that it his honor. And the early Chachamim, among them were those who split wood that was used to cook with, or to salt the meats, or lengthen {?} the wicks or light candles, and among them were those who went out and shopped for things needed for Shabbos, for eating and drinking, even though it was not their general way to do so. And whoever increases in this matter, he is praised.

(Shabbat 119) And he should not say 'this reduces my honor', for this is his honor, that he honors the Shabbat.

And he wrote this in this language of obligation. And the Tur and Shulchan Aruch did not utilize the language of obligation, see there. And even he (Rambam), his intent is not to absolute obligation, for behold many, and great ones included, do not do by themselves. And further, in Sha's there it deems as distinguished those who do this by their own actions. We deduce from this that the multitudes did not do so by themselves, but rather it is a hiddur mitzvah. And by us, our wives do this, and prepare by themselves for the honor of Shabbos, and ishto kegufo."

One of the mitzvos specially singled out by Chazal for women was being mafrish challah, the special portion historically for kohanim from any dough, not baking challah, a certain stylized break which become customary to prepare for Shabbos in recent times. It is a homonym of challah / challah. And these three were singled out because, as the one usually in charge of these, she could cause serve as a tremendous stumbling block. See Shabbat 31a.