You can split the string to get an array, sort them by passing each string to the Date constructor, and then join it back together.

let dates = "6/11/2015, 6/1/2015, 6/22/2015, 6/7/2015, 5/11/2015";
let res = dates.split(", ").sort((a,b)=>new Date(a) - new Date(b)).join(", ");
console.log(res);

The way I think, this could be done easily using split() to extract each day, month, year, and then, use those values to construct Dates objects. Finally, you compare and sort those dates.

const splitDatesByComma = dates.split(',').map((el) => el.trim())
            
const dates = splitDatesByComma.map((el) => {
  const splitDate = el.split('/')
                
  // Create a Date for splitted string. Month is 0 based
  return new Date(splitDate[2], splitDate[1] - 1, splitDate[0], 0, 0, 0, 0)
})
            
const sortDatesDescending = dates.sort((dateA, dateB) => {
  if (dateA > dateB) {
    return -1
  }
  if (dateA < dateB) {
    return 1
  }
    return 0
  })

// Format sorted dates to string and join them.
const datesFormattedAndSorted = sortDatesDescending.map((date) => {
  return `${date.getDate()}/${date.getMonth() + 1}/${date.getFullYear()}`
})
console.log(datesFormattedAndSorted.join(', '))

I've done this at CodePen with Vue if anyone is interested: https://codepen.io/LucasFer/pen/mdMmqrN