Puncetto (also punchetto, punto alpino, punto avorio, Saracen point,etc.): needle-knotted lace made using valsesian st and bar st. [Stillwell];  made in Valsesia, Piedmont district, 16th cent. to now (?), fine thread, geometric classical designs, not made commercially [Eveleth];  produced in 19th and 20th cents. in the mountain valleys of Tuscany, may be a descendant of punto a groppo, though much lighter in texture and more open in design [Earnshaw]. (Tess)

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How about "Puncetto"? There's a section on it in the Anchor Manual of Needlework -- from Valsesia in Italy, says it's often worked in the Piedmont and the Alps, especially on the national costume. It looks like a nice, solid, long-wearing button-hole stitch decoration for clothing and household textiles.

Would I call it lace? Well, it has holes. . . but I'm inclined to see a middle ground of decorative needlework that sits between "absolutely lace" and "absolutely embroidery of some form on solid cloth." In things like Hardanger, it starts with cloth and ends up with decoration and holes; in things like Puncetto, it starts with nothing and makes the "cloth" as it goes along, leaving the holes. Maybe it depends on what the particular area had available when the technique developed - more cloth or more thread.
Su Carter

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I couldn't find Valsesia in my atlas but an internet search
yielded Alagna Valsesia - a ski resort in the Italian alps.
This then makes it likely that the lace is the needlelace
described as Puncetto in 'Golden Hands' -(75 weekly issues
on crafts; the set I picked up in a seconhand shop was published
by Marshall Cavendish Ltd, London,in 1972).  To quote:

'Alpine girls with plenty of time to spare use fine yarn and produce delicate results like the traditional cloth shown here [more delicate and open than the illustration on the book - J].  This particular form of needle-made lace is called Puncetto
(pronounced poon-che-toe) and comes from a word in the dialect
of Northern Italy meaning stitch.'

The very simple modern pattern given is a pyramid edging consisting of a series of triangles made using twisted button hole stitch.

I think the type of lace that you are referring to is called Puncetto.  It is a name used for Oriental lace and this is a needlelace which resembles Tunisian crotchet.  In contrast to Armenien lace, where the knots are sewn only from left to right - the knots in Oriental lace are sewn both in the "forward and back rows".  The lace is used for bedlinen, blouses and
underwear.  It is made in the Italian Alps, in the Valsésia Valley at the foot of Monte Rosa and is also known as Punto alpino or Punto avorio.

This info is to be found in one of the best reference books I have ever come across - the Danish "Bonniers encyclopædia of handcraft".  Such a pity that it isn't translated to English.  It is in 20 volumes with over 2500 articles by leading international experts and more than 3000 (excellent) colour photos.
best wishes and keep bobbin' along from Avril Bayne in Denmark
 

Hold the material in the left hand; working over a selvedge or
narrow hem:
'Working from left to right, bring the needle upward under the
edge of the cloth, 2 or 3 threads in from the edge. Take the
working thread in the right hand and wind it round the needle
once.  (Take it from left to right and back, passing it in
front of the needle first.)  Pull the needle up, tightening
the knot you have made.'
The second row is worked from right to left, the needle being
brought up between two knots of the preceeding row, the thread
being wound round the needle from right to left, passing  it
in front of the needle first.
Jay