My inclination is to call this reaction a "phosphono-Claisen condensation." A Claisen condensation is the alpha-acylation of an ester with another ester. In this case, the typical carbon-centered ester is replaced with a phosphonate, so phosphono-Claisen condensation specifies that difference. Avoiding the proper name, "alpha-acylation of a phosphonate" is a suitable description of the reaction.

Montchamp and coworkers published a paper describing the "phospha-Claisen" reaction, which they limit to the condensation of two phosphonates to give a 1,1-diphosphono compound.

The website maintained by the venerable Hans Reich describes this reaction as "Claisen-like":

(MeO)2P(O)CH2Li will also react with carboxylic acid derivatives like esters, lactones and amides in a Claisen-like reaction to form keto phosphonates.

The Organic Chemistry Portal describes the reaction as "the condensation of esters and phosphonates."

A brief survey of the literature featuring this reaction type (Thanks SciFinder!) doesn't show any convention for naming this reaction beyond calling it the acylation of a phosphonate. In my SciFinder searching, the earliest usage of this reaction is from 1970, and that paper cites a Corey paper from 1966. That paper doesn't have a proper experimental on the reaction, but contains the note:

Anions derived from 1 (X = H or alkyl) are of value in the synthesis of keto phosphonates 1 (X = RCO) by reaction with esters. The preparation of these intermediates, which are of value in olefin synthesis [see H. Takahashi, K. Fujiwara, and M. Ohta, Bull. Chem. Soc. Japan, 35, 1498 (1962)], is usually impractical by the Michaelis-Arbusov reaction.

It's a little nebulous who performed this chemistry first, but perhaps someone could have christened it the "Corey-Claisen condensation" or the "Corey variant of the Claisen condensation". Of course there is already a Corey-Claisen rearrangement so this would be really confusing.

My inclination is to call this reaction a "phosphono-Claisen condensation." A Claisen condensation is the alpha-acylation of an ester with another ester. In this case, the typical carbon-centered ester is replaced with a phosphonate, so phosphono-Claisen condensation specifies that difference. Avoiding the proper name, "alpha-acylation of a phosphonate" is a suitable description of the reaction.

Montchamp and coworkers published a paper describing the "phospha-Claisen condensation" reaction, which they limit to the condensation of two phosphonates to give a 1,1-diphosphono compound.

The website maintained by the venerable Hans Reich describes this reaction as "Claisen-like":

(MeO)2P(O)CH2Li will also react with carboxylic acid derivatives like esters, lactones and amides in a Claisen-like reaction to form keto phosphonates.

The Organic Chemistry Portal describes the reaction as "the condensation of esters and phosphonates."

A brief survey of the literature featuring this reaction type (Thanks SciFinder!) doesn't show any convention for naming this reaction beyond calling it the acylation of a phosphonate. In my SciFinder searching, the earliest usage of this reaction is from 1970, and that paper cites a Corey paper from 1966. The latter paper doesn't have a proper experimental on the reaction but contains the note:

Anions derived from 1 (X = H or alkyl) are of value in the synthesis of keto phosphonates 1 (X = RCO) by reaction with esters. The preparation of these intermediates, which are of value in olefin synthesis [see H. Takahashi, K. Fujiwara, and M. Ohta, Bull. Chem. Soc. Japan, 35, 1498 (1962)], is usually impractical by the Michaelis-Arbusov reaction.

It's a little nebulous who performed this chemistry first, but perhaps someone could have christened it the "Corey-Claisen condensation" or the "Corey variant of the Claisen condensation". Of course there is already a Corey-Claisen rearrangement so this would be really confusing.