These two pieces for piano were probably composed in 1912 at La Grangette, the country home of the Godebskis; and they are dedicated to Ida and Cipa Godebski. They were first performed at the Salle Pleyel on 10 December 1913 by Alfred Casella.
They are pastiches of two composers whom Ravel had admired since his student days. À la manière de Borodine is an elegant waltz. À la manière de Chabrier is equally brief but more complex, in that the theme that it uses is actually by Gounod (Siebel's song "Faîtes-lui mes aveux", from Faust), but treated as Chabrier might have adapted it.
On Chabrier's importance for Ravel, Roger Delage has published an article: "Ravel and Chabrier", in The Musical Quarterly, vol.61, no.4, (Oct.1975), pp. 546-552. [For those with access to JSTOR, an online version of this article is available.]
There are also some useful observations on Ravel and Chabrier by Barbara L. Kelly in her chapter "History and homage", in The Cambridge Companion to Ravel, , pp. 12-14.