A "lassonier" is an element of the formwork used for rammed earth wall construction. It is specifically one of the lower horizontal cross-pieces of the wooden box into which the earth is packed. This is clearly illustrated in the following very useful source, a book called La Ferme moderne: traité de construction rurale, by an engineer called M. Abadie (Paris: Larousse, undated but bookplate of 1894). See the illustration on p. 54 of the book (p. 58 of the file):
http://storage.lib.uchicago.edu/pres/2013/pres2013-0059.pdf

Then compare it with this illustration, showing formwork labelled in English. about halfway down. (The page on which this illustration appears won't open for me.) As you can see, the corresponding member is called a transverse tie here:
http://www.nzdl.org/gsdl/collect/cdl/archives/HASH5149/5f314...

"Cross tie" could also be used:

"The presence of the putlog holes of the wooden cross ties helps us to reconstruct the process of the wall's construction."
Rammed Earth Conservation, p. 110a
https://books.google.es/books?id=A41rcHptt38C&pg=PA110&lpg=P...

The upper horizontal ties, according to these sources, are called traverse or trévire in French.

Here's a very interesting document, an account of the trade by a French rammed earth wall builder: "Témoignage de Philibert Perraud, né le 29 juillet 1923, maçon piseur à la retraite à Bey (01), recueilli par Nicolas Meunier en aout 2010". It has an English translation too. "Lasonnier" is translated "putlog" on p. 3 (top), but I think it's better to keep that word for "boulin".

You may be interested in this Spanish-English question I answered on the subject last year:
http://www.proz.com/kudoz/5789966

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Note added at 3 hrs (2016-02-10 18:13:34 GMT)
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Sorry; I forgot to give the reference for Philibert Perraud's accound, mentioned at the end:
http://www.gpem.ifsttar.fr/fileadmin/redaction/1_institut/1....

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Note added at 4 hrs (2016-02-10 18:22:33 GMT)
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By the way, I have only just looked at the book Liz cited in the discussion area; it confirms what lassoniers are: the ties marked (a) on the illustration. It says that "clefs" is a synonym, so that solves that term in your previous question.