Stir together beef, crackers, Worcestershire, and egg in a bowl. Season with salt and pepper, if desired. Shape mixture into 6 patties.
Cook patties in a large skillet over medium heat to desired degree of doneness. Remove from skillet and keep warm; wipe skillet clean.
Melt butter in same skillet over medium heat.
Add onion and garlic; cook, stirring occasionally, 5 minutes.
Add tomato paste; cook, stirring, 2 minutes.
Add tomatoes and broth; bring to a boil, reduce heat, and simmer.
Return burgers to skillet; cook 1 minute or until heated through.
Sprinkle with basil and thyme.