Here is a more generic solution for any given weekday. Working demo on jsfiddle
var myIsoWeekDay = 2; // say our weeks start on tuesday, for monday you would type 1, etc.
var startOfPeriod = moment("2013-06-23T00:00:00"),
// how many days do we have to substract?
var daysToSubtract = moment(startOfPeriod).isoWeekday() >= myIsoWeekDay ?
moment(startOfPeriod).isoWeekday() - myIsoWeekDay :
7 + moment(startOfPeriod).isoWeekday() - myIsoWeekDay;
// subtract days from start of period
var begin = moment(startOfPeriod).subtract('d', daysToSubtract);