You will need to narrow down the place first. There are no national BMDs and "Normandy" covers five departments (each with separate archives, and most archives don't have name-searchable indexes).
It is possible Gustave married in London and then returned to France
https://familysearch.org/pal:/MM9.1.1/2DR5-MZZGustave Desire Coulette
possible spouse
Emilie Kaetzel
Searching on these names shows this (may need a free geneanet account)
http://gw.geneanet.org/aportaz?lang=fr;p=gustave+desire;n=couletteWhich shows a daughter Madeline b. 17e Paris in 1890.
Paris is split into 20 arrondissements ("Paris" families may also sometimes be found in the suburbs surrounding - these records are then in different archives). The archives for Paris proper are online:
http://canadp-archivesenligne.paris.fr/archives_etat_civil/index.php accédez aux tables décennales gives you access to the "ten year indexes". There will be one set for each arrondissement. Names are in alphabetical order although in some records the definition of "alphabetical" is a little loose.
In the indexes for 17e from 1883 to 1892 I find:
24 April 1890 - birth of Madeline Louise Coulette
also
1 April 1884 - marriage of Gustave Desire (transcription)
Transcriptions of marriages were records for marriages that took place elsewhere.
This one shows that in 1884 Gustave Desire Coulette and Emilie Kaetzel, residing in 17e Paris, presented their original marriage certificate from England and a translation, which shows they married 22 Dec 1881 at the St. Hanover Square register office.
Gustave Desire Coulette, aged 31, a chef (cuisinier), son of the late Gustave Coulette, dealer, residing at 32 Grosvenor Gardens.
Emilie Kaetzel, aged 21, without occupation, daughter of the late Jean Kaetzel, dealer, also residing at 32 Grosvenor Gardens.
Witnesses were a Louise George and "S. Coulette".
Madeline Louise Coulette's birth doesn't reveal much more info as to her father's origins, unfortunately. He does gain an extra first name, "Marie Gustave Desire". (This isn't unusual).
http://www.geneaservice.com - looking on here, it appears that at some point son Gustave Pierre Coulette was on French electoral rolls giving his birthplace as Levallois (Levallois-Perret).
- also a possible reference to a Suzanne Jeanne Coulette b. Chantilly, Oise in 1885 and married in Havre.
Levallois-Perret is one of those Paris suburbs where the records aren't held in "Paris". Hauts-de-Siene is the archive:
http://archives.hauts-de-seine.net/archives/rechercher/archives-en-ligne/(this one is a right sod to use)
Gustave Pierre Coulette was born there 4 August 1886. His father is "Gustave Marie Desire" Coulette.
Finally, Suzanne Jeanne Coulette was born in Chantilly, Oise. Their archives are here:
http://archives.oise.fr/archives-en-ligne/etat-civil/ (will need free account)
Father is just "Gustave Coulette" here. Address possibly Grand Ecuries
Marginal notes on the birth show she married in Havre in 1913 to Pierre Mignon and died in Rouen in 1963.
The family are still there on the census of 1886, page 4, although this doesn't show much (no birthplaces on this census) - just that Suzanne was the only child in the household. They were not at the Chateau
For her marriage we need to go to yet another archive site:
http://recherche.archivesdepartementales76.net/Luckily, the marriages are online for 1913 (quite a few sites have nothing past 1912)
At her marriage, Suzanne's parents are listed as Gustave Coulette, Maître d'hôtel, and Emilie Kaeztel, both residing in London, both present at the marriage.
I cannot see Gustave or Emilie's deaths in the UK - if they returned to France and died there, there's a chance that those records will give information about Gustave's parents and place of birth (if the person registering the death knew them).