Salvador Dalí: 
"As a little boy at school, I stole an old slipper belonging to the teacher, and used it as a hat in the games I played in solitude. In 1936, I constructed a Surrealist object with an old slipper of Gala’s and a glass of warm milk. Years after my school-boy prank, a photo of Gala crowned by the cupolas of Saint Basil revived my early fantasy of the “slipper-hat.” Finally Madame Schiaparelli launched the famous slipper-hat. Gala wore it first; and Mrs. Fellowes appeared in it during the summer, at Venice. 
From: The Secret Life Of Salvador Dalí (Dial Press, 1942)

1937 | Elsa Schiaparelli shoe-hat
Drawing by Marcel Vertès
Source: Archivio Alinari

Dalí's statement partly contradicts the beginning of the shoe-hat description from the Palais Galliera website:

The high-heeled Schiaparelli shoe hat was created in 1937 in collaboration with Salvador Dalí. The idea came from a photograph taken in 1933 by Dali's wife Gala, showing the artist wearing a woman's shoe on his head and another one on his shoulder.  [*]

1932 (According to DaliPhoto.com) | Salvador Dalí
Photograph by Gala Dalí
Source: DaliPhoto.com (Apparently not available anymore)

The hat was captured for posterity in a photography by Georges Saad, published in the October 1937 number of L’Officiel de la Mode et de la Couture … [*]

1937, October | L'Officiel De La Mode # 194
Caption reads: the very original hat represents a black felt shoe with red heel
Photograph by George Saad
Source: L'Officiel De La Mode

... and Gala herself can be seen wearing it in a photo taken by André Maillet in 1938.  [*]

1938 | Gala Dalí wearing the Schiaparelli outfit shown above
Photograph by André Maillet

This hat, which belonged to Gala, entered the museum collection [Palais Galliera] in 2013. On her return to France from the United States in 1947, Gala gave it to her daughter Cécile Eluard. [*]

1937 | Schiaparelli shoe-hat at the Palais Galliera
Musée de la Mode de la Ville de Paris

Two other examples of this emblematic accessory are known: one with a red heel in the Andrea Pfister Collection, the other with a black heel in the Metropolitan Museum of Art in New York  [*]
Source: Palais Galliera [*]

1937 | Schiaparelli shoe-hat at the MET, NY
Marking: [label] "Schiaparelli, Winter, 1937-38"

And now to wrap it up, after the abused "glass of warm milk" above, more words of wisdom from the Maestro:

All my life I have been preoccupied with shoes, which I have utilized in several surrealist objects and pictures, to the point of making a kind of divinity of them. In 1936 I went so far as to put shoes on heads; and Elsa Schiaparelli created a hat after my idea. Daisy Fellowes appeared in Venice with this shoe-hat on her head. The shoe, in fact, appears to me to be the object most charged with realistic virtues as opposed to musical objects which I have always tried to represent as demolished, crushed, soft–cellos of rotten meat, etc. One of my latest pictures represents a pair of shoes. I spent two long months copying them from a model, and I worked over them with the same love and the same objectivity as Raphael painting a Madonna. 
Salvador Dalí
The Secret Life Of Salvador Dalí (Dial Press, 1942)

Last but not least, the André Perugia shoe used by Elsa Schiaparelli to model her shoe-hat.

Black suede lace-up shoe for Schiaparelli
Photograph by Cynthia Hampton
Source: Heavenly Soles (Abbeville Press, NY, 1989)

The Origin of the red sole

1937 | Schiaparelli sketches for the shoe-hat
Pen on paper | Musée des Arts de la Mode, Paris
Source: Heavenly Soles (Abbeville Press, NY, 1989)


Assorted VanDALIsm

1985 | Katherine Helmond 
In Brazil | Written/directed by Terry Gilliam