According to the Texas Department of Public Safety, this happened on Tuesday between 1 and 1:30 a.m. at a church on 13th and Lavaca streets in downtown Austin, just outside of Capitol grounds.
Investigators said the man threw the cocktail at the church, but by the time DPS troopers arrived, he was gone.
These type of bottle-based incendiary devices are typically filled with flammable liquids.
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Two photos of the man were released Friday afternoon. He was seen wearing black shorts, white sneakers and a white, long-sleeved shirt with a design featuring the words "Pizza Planet," which refers to a fictional restaurant from the popular Disney movie "Toy Story."
The man was also seen wearing a black backpack and latex gloves and was holding a skateboard, according to investigators.
It's unclear if the church sustained any damage. An arson investigator with the Austin Fire Department is helping DPS with the investigation.
Texas Crime Stoppers is offering up to $10,000 for information that leads to the identification and arrest of the man.
To be eligible for cash rewards, you must provide information to authorities by calling the Crime Stoppers hotline at 1-800-252-TIPS (8477) or submit a tip online.
All tips are anonymous, regardless of how they are submitted.