Melt butter in a 1-quart saucepan set over medium heat.
Add flour to pan and stir to incorporate and form a roux. Continue to stir until roux becomes dark brown, 6 to 8 minutes.
Add shallots, garlic, and thyme and continue to cook, stirring often, until shallots and garlic are fragrant, about 1 minute.
Pour Worcestershire sauce into pan and cook until nearly evaporated, about 1 minute.
Add beef broth, season with salt and pepper, bring to a boil, reduce to a simmer, and continue to cook until sauce has reduced and no floury taste remains, about 20 minutes.
Remove from heat. Serve immediately or set aside and reheat when ready to use.