The below works:
using Test
anon = x::Int -> x^2
(f::typeof(anon))(x::Float64) = x *2.
@test anon(3) == 9
@test anon(3.) == 6.0
Although why use anonymous function if you are going to end up writing the second line? By writing a second line, you are no longer achieving brevity by using anonymous functions. And by assigning the anonymous function to a variable, you are already polluting the namespace.