The Godfather, an enduring classic from the 1970s, is basically a Scotch whisky-based Old Fashioned sweetened with flavoursome liqueur rather than relatively flavourless sugar.

It's the matriarch of a whole "God" damn family of similar cocktails:
Godfather: Scotch whisky and amaretto
Godmother: vodka and amaretto
Godchild: brandy and amaretto

What's a Godfather called when made with Bourbon rather than Scotch? Well, according to Stanley M. Jones' 1977 Jones' Complete Barguide (page 233), it's simply The Boss:

It's not just the base-spirit that can be swapped to produce variations on the classic Godfather, liqueurs and other sweeteners besides just amaretto can be used, along with aromatic bitters.

Almond Old Fashioned (bourbon-based)
With: Bourbon, amaretto, dark sugar syrup, aromatic bitters.
We say: Bourbon's sweet notes are lifted by amaretto's almond richness.

Almond Old Fashioned (tequila-based)
With: Reposado tequila, amaretto liqueur, agave syrup and orange bitters.
We say: Just as it says on the tin. Yumm.

Brooklyn Godfather
With: Bourbon, amaretto liqueur, dry vermouth, and rosso vermouth.
We say: An amaretto Brooklyn or a bourbon-based Godfather with vermouth.

Big Boss
With: Bottled-in-Bond rye whiskey, amaretto and bitters.
We say: A Godfather made with bourbon is "The Boss" so it stands to reason that with bottled-in-bond rye it is the Big Boss.

Cicada
With: Bourbon, amaretto liqueur, cream and sugar syrup.
We say: A creamy Godfather.

Explorer 'Martini'
With: Scotch whisky, amaretto liqueur, and amontillado sherry.
We say: Add a splash of sherry to a Godfather and you've an explorer.

Fat Godfather
With: Fat-washed Scotch whisky, amaretto liqueur and whiskey barrel bitters.
We say: This Godfather has been on the bacon.

French Connection
With: Cognac and amaretto liqueur.
We say: A Godchild but traditionally specifically made with cognac rather than brandy and served in a snifter glass without ice.

Fruity Godchild
With: Bourbon, rye whiskey, amaretto liqueur, crème de noyau, maraschino liqueur and aromatic bitters.
We say: A rich fruity riff on a Godfather with a surprisingly dry finish.

Godfather Sour
With: Scotch whisky. Amaretto liqueur, lemon juice, sugar syrup and egg white.
We say: A rich almondy whisky sour.

Jack's Godfather Highball
With: Tennessee whiskey, amaretto liqueur and cola.
We say: The ultimate Italian American highball.

Loch Almond Highball
With: Scotch whisky, amaretto, liqueur and ginger ale.
We say: Add ginger ale to lengthen a Godfather and we've a Loch Almond Highball.

Madurised Old Fashioned Godfather
With: Bourbon, rye whiskey, Scotch whisky, amaretto liqueur, and Verdelho madeira.
We say: A trio of whiskies with rich vinious nutty complexity.

Smoked Old Fashioned Godfather
With: Bourbon, rye whiskey, Scotch whisky, amaretto liqueur and Islay peated single malt.
We say: A quartet of whiskies with rich almond notes.

Stiletto Manhattan
With: Tennessee whiskey, amaretto, dry vermouth, orange bitters.
We say: Add a splash of vermouth to a Godfather, and you've a Manhattan.

Toasted Old Fashioned Godfather
With: Rye whiskey, bourbon whiskey, amaretto liqueur and aromatic bitters.
We say: A complex, dry and boozy late-night sipper.