fitting - Is there a way to produce a FittedModel object as the solution to a regularized regression?

I can get the Ridge or Tikhonov regression solution to Y = \[Beta].X easily enough as

\[Beta] = 
 Inverse[Transpose[X].X + \[Lambda] IdentityMatrix[
      Dimensions[X][[2]]]].Transpose[X].Y

for a given regularization parameter \[Lambda].

But I want to use LinearFitModel (or otherwise) to create a FittedModel object so I have access to all it's availible statistics and properties. I haven't been able to find any documentation on how to do this or if it's even possible. Is there a way to do this? Thanks!