Grind garlic and salt together with a mortar and pestle. Add anchovies; grind with mortar and pestle until a coarse paste forms.
Heat olive oil in a skillet over medium heat. Whisk butter into oil until butter melts and foams, about 30 seconds. Reduce heat to low and stir garlic-anchovy mixture into oil mixture; cook and stir until flavors blend and mixture is hot, about 5 minutes. Remove from heat and stir red wine vinegar and red pepper flakes into oil mixture.