No birth record, even as ‘Edward Alfred’ on its own in FreeBMD.

On 19th April 1877, while curate of Hawley, Hants, he assisted his elder brother Hugh at the marriage of their cousin Mary Hodgson Gillett Stedman to Rev Henry Eden Trotter (born Brown).
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In 1881, living at Willage Street Rectory, Woolsthorpe, Lincolns:

Surname  Forename  Age  Relation  Occupation  Birth Place

Gillett Edward A 38 Head M A Rector Of Woolsthorpe  Waltham, Leicestershire
Gillett Catherine E 36 Wife Scotland
Gillett George G 7 Son Scholar Hawley, Hampshire  
Gillett Walter S 5 Son Scholar Hawley, Hampshire  
Gillett Edward S 3 Son Hawley, Hampshire  
Gillett Charles 3M Son Woolsthorpe, Lincolnshire
Jerome Eliza A 58 Servant Domestic Servant Cook  Egham, Surrey
Lock Sarah A 21 Servant Domestic Servant Housemaid  Waltham, Leicestershire
Croucher Maria 30 Servant Domestic Servant Nurse Cosham, Hampshire
Davis Emily 17 Servant Domestic Servant Housemaid  Sippingham, Berkshire
Matthews Sarah A 21 Servant Domestic Servant Nurse  Waltham, Leicestershire

Ref: RG11/3230/F?

Address: Rectory
Parish: Woolsthorpe
Registration District: Gratham
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In 1901 he had no family with him but said he was married (not widowed), had 4 servants, and was still a clergyman at Woolthorpe, Lincs.
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A possible death:

Surname   First name(s)   Age   District   Vol  Page
Deaths Dec 1926   (>99%)
Gillett Edward A 84 Bicester 3a 1177
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Possibly they m. in Scotland, no record on FreeBMD.