What is double titration?
Double titration also often refers to Gilson's titration method (also known as &quot;Gilman color tests&quot; or &quot;Gilman double titration&quot;) for determining the concentration of organometallic compounds in hydrocarbon solvents.
At the first stage the total concentration of the base in the solution is determined.
At the second stage the residual amount of alkali after the organometallic compound reacts with the halide is established, allowing to calculate the concentration of metalloorganic compound. [1]

For the double titrations, a 3 ml aliquot of the solution to be analyzed was
withdrawn by pipette using a suction bulb and added to 10 ml of ether.
The solutionwas hydrolyzed with 10 ml of distilled water and titrated with standard acid, using phenolphthalein as indicator, to give the total present.
A second 3 ml aliquot was withdrawn and added 10 of ether, containing 1 ml of the halide, under an atmosphere of nitrogen.
The solution was swirled gently and allowed to stand for 2 min, after which it was hydrolyzed with distilled water and titrated immediately with standard acid, again using phenolphthalein as indicator.
In titrating the latter solution the end point is easily overstepped, since the
aqueous layer decolorizes before the ether layer. Vigorous shaking near the end point
is recommended.
There is the possibility of interference from hydrolysis of the halide, so that the titration should be carried out within a reasonable length of time.

An updated cumulative summary for the method can be found in Wakefield's Organolithium Methods [2, pp. 17–18]:

For detecting the presence of an organolithium compound (for example for checking whether any remains at the finish of a reaction) Gilman's Colour Test 1 is convenient:
$$
\begin{align}
\ce{(4-Me2NC6H4)2C=O + RLi &amp;-&gt; (4-Me2NC6H4)2C(R)OLi}\\
\ce{ &amp;-&gt;[(i) H2O][(ii) I2, AcOH] [(4-Me2NC6H4)2CR]+ I-}
\end{align}
$$
A sample (0.5—1 ml) of the test solution is added to an equal volume of a I solution of Michler's ketone, 4.4'-bis(dimethylamino)benzophenone, in dry benzene.
A few drops of water are added, and the mixture is shaken.*
A few drops of an 0.2% solution of iodine in glacial acetic acid are added,
resulting in the formation of a blue or green colour.
Gaidis proposed two modifications, to distinguish between alkyl- and
aryllithium compounds (15).
(a) Following hydrolysis, the solution is buffered to pH 9 by the addition of 20% aqueous catechol (1—2 ml).
A few drops Of iodine in benzene are added and the solution is shaken.
A green to blue colour in the organic layer indicates the presence of an aryllithium compound.
The mixture is then acidified with acetic acid to reveal the presence Of an alkyllithium compound.
(b) Following hydrolysis the solution is acidified with acetic acid (5—15
drops).
20% aqueous sodium bisulphite is added; aryl- but not alkyllithium
compounds give a green colour.
Gilman's Colour Test 2 (16) which also distinguishes alkyl- from aryllithium compounds, depends on the rapid metal—halogen exchange of the former with 4-bromo-N,N-dimethylaniline:
$$
\begin{align}
\ce{4-BrC6H4NMe2 + RLi &amp;&lt;=&gt;&gt; 4-Li2C6H4NMe2 + RBr}\\
\ce{4-Li2C6H4NMe2 &amp;-&gt;[(i) Ph2CO][(ii) H3O+] [Ph2CC6H4NMe2]+}
\end{align}
$$
The solution to be tested (0.5-1 ml) is added to an equal volume of a solution of 4-bromo-N,N-dimethylaniline in dry benzene.
A 15% solution of benzophenone in dry benzene (1 ml) is added.
After a few seconds, water is added and then the mixture is acidified with concentrated hydrochloric acid.
A positive result is indicated by a red colour in the aqueous layer.

* Aerial oxidation sometimes leads to formation of the blue-green at this stage.

References

Gilman, H.; Cartledge, F. K.; Sim, S.-Y. The Analysis of Organic-Substituted Group IVB Lithium Compounds. Journal of Organometallic Chemistry 1963, 1 (1), 8–14. DOI: 10.1016/S0022-328X(00)80039-X.
Wakefield, B. J. Organolithium Methods; Best synthetic methods; Academic Press: London; San Diego, 1988. ISBN 978-0-12-730940-8.