How is chlorocarbene synthetised?
The formation of dihalocarbenes (at least cloro- and bromo-counterparts) in reactions of haloforms with bases has been well established (e.g., Ref.1,2). In 1959, Closs and Closs introduced a novel reaction to prepare chlorocarbene as a reaction intermediate in situ in order to synthesize chlorocyclopropanes derivatives (Ref.3). They have reported that when n-butyllithium is added to methylene chloride in the presence of olefins at $\pu{—25 °C}$, chlorocyclopropanes are formed in fair to good yields depending on the structure of the olefin. They further claimed that the novel procedure strongly suggests the existence of chlorocarbene as a reaction intermediate. 

They further developed the method and published a full paper as a general method (Ref.4) where abstract states:


  Alkyllithium compounds react with methylene chloride in the presence of olefins to form chlorocyclopropanes. This reaction is believed to proceed through the intermediate chlorocarbene. Seven alkyl-substituted chlorocyclopropanes have been prepared by this method. Configurations are assigned to the products by considerations of predictible non-bonded interactions in the transition states of the addition of chlorocarbene to the olefins. The configurations are confirmed by n.m.r. spectra.


Later, Billups and Haley have used the same method to synthesize spiropentadiene (Ref.5).

References:


W. von E. Doering, A. Kentaro Hoffmann, “The Addition of Dichlorocarbene to Olefins,” J. Am. Chem. Soc. 1954, 76(23), 6162–6165 (https://doi.org/10.1021/ja01652a087).
W. von E. Doering, Wm. A. Henderson, Jr., “The Electron-seeking Demands of Dichlorocarbene in its Addition to Olefins,” J. Am. Chem. Soc. 1958, 80(19), 5274–5277 (https://doi.org/10.1021/ja01552a065).
Gerhard L. Closs, Lisellote E. Closs, “Synthesis of Chlorocyclopropanes from Methylene Chloride and Olefins,” J. Am. Chem. Soc. 1959, 81(18), 4996–4997 (https://doi.org/10.1021/ja01527a057).
Gerhard L. Closs, Lisellote E. Closs, “Carbenes from Alkyl Halides and Organolithium Compounds. I. Synthesis of Chlorocyclopropanes,” J. Am. Chem. Soc. 1960, 82(21), 5723–5728 (https://doi.org/10.1021/ja01506a041).
W. E. Billups, Michael M. Haley, “Spiropentadiene,” J. Am. Chem. Soc. 1991, 113(13), 5084–5085 (https://doi.org/10.1021/ja00013a067).