Class-E-to-surface "extensions" are not "within the lateral boundaries of controlled airspace designated to the surface for an airport", so the requirement of FAR 91.155 "Except as provided in §91.157, no person may operate an aircraft beneath the ceiling under VFR within the lateral boundaries of controlled airspace designated to the surface for an airport when the ceiling is less than 1,000 feet." does not apply.
The key phrase being "designated for an airport". The first link given below explores why this is significant.
Similarly, it would not be legal to operate with less than the standard class E cloud clearance and visibility requirements within the "extension", even if a Special VFR clearance has been granted. FAR 91.157 only allows for Special VFR operations "below 10,000 feet MSL within the airspace contained by the upward extension of the lateral boundaries of the controlled airspace designated to the surface for an airport."
Thus the only practical purpose of a Class-E-to-surface "extension" is to extend standard Class E cloud clearance and visibility requirements all the way to the surface, for the protection of IFR traffic emerging from clouds at low altitudes. It would not be safe to have a VFR aircraft skimming just over or under a cloud deck in this area, as would be perfectly legal in Class G airspace.
Once a VFR aircraft is above the level of the floor of the Class E airspace surrounding a Class-E-to-surface "extension" (typically 700' AGL), there is no practical significance as to whether the aircraft is within the "lateral boundaries" of the "extension" or not. Of course, this is not true for other kind of Class-E-to-surface airspace-- the kind that actually surrounds the airport whose approaches it protects. I.e. the kind that is "designated for an airport".
For much more on the precise meaning of "within the lateral boundaries of controlled airspace designated to the surface for an airport", please see Which parts of class E airspace can an ultralight (part 103) fly in without prior ATC authorization? . This link should link directly to the answer starting with the title "Shortest Answer" -- which, naturally, is actually the longest answer. The precise wording that this link addresses is "within the lateral boundaries of the surface area of Class E airspace designated for an airport", which is extremely similar to the wording in 91.155 and 91.157. Central to that answer is this FAA Memorandum from January 10, 2018, addressing the operation of "Small Unmanned Aircraft" (drones) in Class-E-to-surface "extensions".
It is likely that we will see some FAA rulings that impact on these topics in the near future. As things stand right now, there appears to be a loophole in regulations that will go into effect in the near future (HR 302) regarding Small Unmanned Aircraft (radio-controlled airplanes and "drones"), which will allow SUA's to operate above 400' altitude without "prior authorization from Air Traffic Control" ONLY if they are in Class-E-to-surface "extensions", and nowhere else in the nation's airspace-- except perhaps in Class E not-to-surface airspace that was entered from a Class-E-to-surface "extension", or after climbing above the top of the Class E floor while flying "with prior authorization" in Class B, C, or D airspace or while flying with "prior authorization" in Class-E-to-surface airspace of the "designated for an airport" variety. It is unlikely that this is what was intended. There are at least two logical ways to fix the problem: 1) extend the 400' AGL limit (in the absence of "prior authorization") to all airspace, not just Class G, or 2) explicitly include all Class-E-to-surface airspace including "extensions" in the airspace in which "prior authorization" is needed. However, it is possible that the FAA will simply bludgeon the issue into submission and overrule the January 10 2018 memorandum, however well-founded it may be, and by ruling that the meaning of "within the lateral boundaries of the surface area of Class E airspace designated for an airport" DOES include the Class-E-to-surface extensions.
