How to synthesise N,N,3‐trimethylbutanamide from 1‐chloro‐2‐methylpropane?
Phosgene, $\ce{O=CCl2}$, is a potent toxin and in no way would I consider it a safe or even standard laboratory reagent (at least on an undergraduate level).

Allow me to lead you down the garden path of organic synthesis.

We have a haloalkane. What do we know about such compounds; they have a polarised $\ce{C-X}$ bond ($\ce{x}$ is a halogen) and are prone to nucleophilic attack/substitution. In this case, we have a nice primary, haloalkane, hence, any nucleophilic substitution is likely to be SN2.

Now, the target is an amide, but what do we note is different about the target versus the starting material (SM)? There's an extra carbon present (and yes there's no halogen).

Warning bells should be going off at this time. What nucleophilic, reagent do we know of that could extend a carbon chain?
I can think of one: $\ce{CN-}$ (cyanide ion).
So, first react with $\ce{KCN}$, second react with conc. $\ce{H2SO4}$ to give a carboxylic acid, third react with $\ce{NH(Me)2}$ to give the product. 
Now, once you've isolated the carboxylic acid, you can react it 'neat' with $\ce{NH(Me)2}$ and molecular sieves. The molecular sieves simply ensure a dry reaction and very high temperatures are needed (approx $\pu{160 ^\circ C}$). You could convert the carboxylic acid to an acyl chloride and react the amine to ensure a very high yield and 'fast' albeit more 'vigorous' reaction; in this case any ordinary organic solvent should suffice.