This seems to be generally left in French with an explanation. I suspect that there is a bit missing after "à châssis", such as "en bois doré"? If so then you need to look again at how you analyse the sentence.

"Chair backs generally rose only to shoulder height, and exception was made for armchairs. If a chair’s back was flat, it was said to be a la reine; ..."
thefrenchfurniturecompany.co.uk/WebsitePages/history.php

"François-Honoré will chair nevertheless the destinies of the family ... Superb pair of giltwood armchairs with flat back called "à la Reine" of Louis XVI ..."
en.artquid.com/rss/16/seat-sofa.html
"Faisant partie d'un ensemble de quatre meubles, ce "fauteuil est dit "fauteuil à la reine" car il présente un dossier plat qui le différencie des sièges dit "en cabriolet" au dossier incurvé."
http://www.framemuseums.org/jsp/fiche_oeuvre.jsp?STNAV=&RUBN...


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Note added at 17 hrs (2008-06-02 10:30:44 GMT)
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Both flat back and cabriolet back chairs were made at the same period, so naming which Louis it was would seem insufficient:

"Faisant partie d'un ensemble de quatre meubles, ce "fauteuil est dit "fauteuil à la reine" car il présente un dossier plat qui le différencie des sièges dit "en cabriolet" au dossier incurvé. Ces deux familles de meubles se distinguent par leur usage : les sièges à la reine ont vocation à être statiques, disposés le long des murs et des boiseries, et sont qualifiés de meublants alors que les sièges en cabriolet, plus légers, se déplacent dans les pièces au gré des usages, gagnant ainsi leur appellation de meubles volants. "
http://www.framemuseums.org/jsp/fiche_oeuvre.jsp?STNAV=&RUBN...




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Note added at 17 hrs (2008-06-02 10:35:32 GMT)
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A reference to Louis XV chairs being made in cabriolet and not just à la reine designs:

"Lots of new pieces of furniture were created in this era, the ‘cabriolet’ chair is one of them. It is a classic Louis XV design. This item will bring light ..."
www.louisandlouis.com/en/-cabriolet-chair-in-louis-xv-desig...