I initially thought this reaction would proceed through an E2 mechanism....
Can you explain why you thought this? I think you have some misunderstandings here - E2 and S
N2 can be ruled out immediately as they require a strong base or good nucleophile, respectively; tertiary alkyl halides are also generally too hindered for S
N2.
E1 and S
N1 both occur by the same
stabilised tertiary carbocation intermediate. This carbocation can either be trapped by MeOH in an S
N1 reaction, or MeOH may abstract a proton in an E1 reaction. Both pathways are viable, but also reversible, ie.
Alkene + MeOH + H
+ <> carbocation + MeOH <> Ether + H
+As MeOH is the solvent, and therefore in
vast excess I expect the eqilibrium to lie on the side of the ether (the S
N1 product).