I always confused eyelet with broderie anglaise (it is a common mistake as broderie anglaise HAS eyelet holes in it) but broderie anglaise is more elaborate and much more time consuming and often features floral patterns.
It became popular in the 19th century in England and may have come or been inspired from the delicate and ornate Ayrshire needlework (Scotland). Broderie anglaise was mostly used on linens, undergarments, and christening gowns.
Sources: 1.] 1922 French fashion plate Pratt Libraries, 2.] Blouse from cherchons.com, 3.] Dahlia Dress from ASOS, 4.] Broderie anglaise cape top 1860-1960.com, 5.] Detail of hem Corsets and Crinolines, 6.] Patterns for broderie anglaise Marquise.de