Blend oil, butter, anchovies and garlic in processor until smooth.
Place mixture in heavy medium saucepan and cook over low heat for 15 minutes, stirring occasionally. (Sauce will separate.) Season to taste with salt and pepper.
Pour sauce into fondue pot or other flameproof casserole and place pot over alcohol burner or gas table burner to keep warm.
Dip the vegetable sticks while holding bread underneath to catch drippings. Eat bread once it is soaked with sauce.