You must have a Refugee Travel Document to return to the United States after temporary travel abroad, unless you have other appropriate documents, such as a Permanent Resident Card and passport.
You must apply for a Refugee Travel Document before you leave the United States. Please note that after the filing of your Form I-131, you will have to attend in person the biometrics services appointment. And if you fail to appear for the appointment, the application may be denied.
Thus, according to the rules you must file Form I-131 and wait till USCIS approves it. And only after you obtain your Refugee Travel Document you are free to travel outside the USA.
Usually, Form I-131 is approved within 3 months, if there are no mistakes and/or omissions in your documents. However, if you need to receive your Refugee Travel Document faster due to an emergency situation, you may be eligible to request an expedited processing.
If you still need to leave the USA before you obtain your Refugee Travel Document or if you want to apply for the travel document outside the USA, we recommend you to consult with an immigration attorney.