Learn English – The etymology оf “handkerchief”

Looking in the OED, I see support for the wine-cask-piercing tool origin, but not for the origin clearly involving one in charge of an auction. There is a connection to wine selling, and this might be close to what you are theorizing.

Etymology: Middle English brocor, -our, brokour, < Anglo-Norman brocour (also broggour) = Old Northern French brokeor ( < Latin type *broccātōrem), nominative brokiere ( < Latin *broccātor) of which Godefroy has one example explained by him as ‘celui qui vend du vin au broc’, as to the precise sense of which see below. The Central French equivalent was brocheor, brochière; and the word is the agent noun of the Old French vb. brochier, Old Northern French brokier ( < Latin *broccāre) in the sense ‘to broach’ or ‘tap’ a cask. Brocheor, brokeor stand in precisely the same relation to the n. broche, broc, and the vb. brochier, brokier, as tapster or rather the earlier tapper stand to the n. tap, and vb. to tap in Teutonic: the brocheor, brokeor, brokour, or broker, was lit. a tapster, who retailed wine ‘from the tap’, and hence, by extension, any retail-dealer, one who bought to sell over again, a second-hand dealer, or who bought for another, hence a jobber, middleman, agent, etc. Compare sense of Latin caupo.

The Romanic vb. broccare was evidently < brocco, brocca in the sense of ‘spike, piercing instrument’ ( < Latin broccus, brocca adj.: see broach n.1). But these nouns appear to have afterwards had their sense modified from the verb, so that in the Old French vendre à broke, or à broche, in modern French vendre à broc, the sense passed from ‘broach’, to ‘broaching, tapping’, and at length to ‘the quantity of wine drawn at a broaching or tapping’, and hence ‘the jug or vessel which held this’, as in modern French broc (from 5 to 10 litres). Anglo-Norman had also a derivative form abrocour, and there were Anglo-Latin words abrocator, abrocamentum; also brocarius ‘proxeneta, interpres et consiliarius contractuum’, and abrocarius. Brocarius appears to have been formed on the n. (broc(c)a, broc(c)us); abrocarius must have been formed on the apparent analogy of brocator, abrocator.

The earliest usages have already lost any connection to wine tapping or selling.

1377 Langland Piers Plowman B. v. 130 Amonges Burgeyses haue I be dwellynge at Londoun, And gert bakbitinge be a brocoure [C. brocor] to blame mennes ware.

1393 Langland Piers Plowman C. vii. 95 Ȝut am ich brocor of bakbytynge · and blame mennes ware.

1582 R. Stanyhurst tr. Virgil First Foure Bookes Æneis i. 14 For gould his carcasse was sold by the broker Achilles.

I would doubt it. Probably the best example of both of the terms is "misfeasance" and "malfeasance". Both attach to the Anglo-French root word "feas-" which is also the root of the modern French "fais" (to do/to make), and usually refer to poor performance in public office (the terms can apply to any job performance though).

However, the terms are distinct at law: "misfeasance" is an unintentional failure to satisfactorily perform the duties of the job, due to incompetence or abdication. "Malfeasance" is an intentional abuse of the powers of the job to cause damage to the office or its constituents.

I cannot think of another use of these two prefixes side by side, but use this rule of thumb: If you want to mean "bad" as in evil, use mal-. If you want to connote "poor" as in "at a low level", use mis-. The loss of a baby in utero due to force majeure or a "poor" uterine environment is a "miscarriage"; it's not a "malcarriage", as the woman did not intentionally use her womb to abuse her unborn child. By contrast, a "malodorous" person or thing literally "smells evil", and figuratively speaking may have a "stink" around their actions or personality. The word is not "misodorous", as most "malodorous" things consciously chose or were specifically designed to smell bad; it's no accident.