
The provided NMR data ($^1$H NMR: $\delta$ 1.17 (d, 6H), 3.02 (m, 1H), 4.10 (s, 2H); $^{13}$C NMR: $\delta$ 17, 37, 39, 210) and molecular formula C$_5$H$_9$BrO suggest that compound [X] is 1-bromo-3-methylbutan-2-one, with the structure (CH$_3$)$_2$CH-CO-CH$_2$Br.
The reaction of an alpha-haloketone like [X] with sodium methoxide (NaOMe) in methanol (MeOH) typically proceeds via the Favorskii rearrangement.
Based on the structure derived from NMR (1-bromo-3-methylbutan-2-one), the expected Favorskii product would be Methyl 3-methylbutanoate.
According to the provided options and correct answer, the major product [Y] is Methyl 3,3-dimethylbutanoate, corresponding to Option B.
This specific product arises from the Favorskii rearrangement of 1-bromo-3,3-dimethylbutan-2-one, (CH$_3$)$_3$C-CO-CH$_2$Br.
Reaction pathway leading to Product [Y]:
$(CH_3)_3C-CO-CH_2Br + MeO^- \rightarrow (CH_3)_3C-CO-CH^-Br + MeOH$
The resulting ester is (CH$_3$)$_3$C-CH$_2$-COOCH$_3$, which is Methyl 3,3-dimethylbutanoate.
Final Answer: The final answer is $\boxed{B}$
For an unambiguous single step synthesis of the following target molecule (TM), the best bond disconnection in its retrosynthetic analysis is

The major products X and Y formed in the following reaction sequences are

For the following reaction, the possible product(s) is/are

The reaction(s) that yield(s) X as the major product is (are) 
The major products E and F in the following reaction sequence are 