We are an expat American family and have been living in France for 3,5 years on our carte de séjour which was provided us by my husband’s company. Since our carte de séjour is expired, the renewal application was made by the company’s lawyer about 2 weeks ago. Can we leave France for a holiday in Turkey and enter back with our American passports while our permits are in the process of being issued?
The EU maintains a document listing residence permits issued by EU and Schengen states along with substitute documents that are formally recognized as equivalent to residence permits. The document is available in PDF format from the website of the EU's department of home affairs.
Under "France," you will find:
Receipts for a renewed application for a resident permit, together with the expired residence permit or ...
Thus, together with the receipt and the expired residence permit, you should be able to travel. (In fact, the receipt itself should mention explicitly that the expiring permit is extended: "dont les effets sont prolongés jusqu'au..." -- "of which the effects are extended until....")
This list is called "Annex 22" because it accompanies the Practical Handbook for Border Guards, also available in PDF format.