A "reaction map" in my lecture handout suggests that the product of this reaction is:
So, basically overall it just merges two ethylacetoacetate molecules together at the central carbon. What is the mechanism for this? My thoughts are to swap the acidic proton for an Iodine atom by making the enol (in acidic conditions to avoid iodoform reaction) and react with $I_2$. Then, the enol of another molecule of EAA substitutes the Iodine to make the product. However, I don't know whether this is $S_N1$ or $S_N2$. Typically alkylation is $S_N2$ (I don't know why, please explain if you can) but in this case its a bulky secondary substituent and the solvent is protic (acidic conditions). Surely it's $S_N1$, right?