If you specifically wish for a projective plane to be contained in a projective CY3 $X$, start with a CY3 $Y$ with a $1/3(1,1,1)$ quotient singularity, and blow up the singular point to get the crepant resolution $f:X\to Y$ with exceptional locus $E\subset X$ isomorphic to the projective plane, with normal bundle being its canonical bundle. There are plenty of such $Y$ among hypersurfaces in weighted projective 4-space and toric fourfolds.
Balazs
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