CompoundPoissonDistribution provides the desired functionality

In[]:=
s=CompoundPoissonDistribution[λ,NormalDistribution[μ,σ]];
In[]:=
{Mean[s],Variance[s]}
Out[]=
{λμ,λ(
2
μ
+
2
σ
)}
In[]:=
Expectation[
2
X
,XNormalDistribution[μ,σ]]
Out[]=
2
μ
+
2
σ