Melt 2 tablespoon butter in the pan over medium heat. Pat dry cod fillets and sprinkle salt and pepper on both sides of the fish.
Dredge cod fillets in all purpose flour.
Sear cod fillets in butter for four minutes on each side. Transfer the cod fillets to a plate.
In the same pan, melt the remaining butter and add shallots and fennel to the pan.
Saute until fennel starts softening. Add in mushrooms and stir well until the vegetables are cooked and soft.
Pour the vegetable broth and heavy cream into the pan. Season carefully with salt and pepper since the vegetable broth has some salt.
Bring to simmer and let the sauce simmer for five to seven minutes.
Place the cod back in the pan and cook for a couple of minutes. Add the lemon juice right before serving.
Serve cod and shallot fennel sauce with dill rice.