Not necessarily milk, but Tibetans were flavoring their tea with Yak butter (po cha) since the Tea Horse Road (Chamagudao - 茶馬古道 ) was established, where Tibetans were trading their sturdy ponies for bricks of dried tea leaves in exchange. Mongolians also have a salty beverage consisting of tea, salt, water and milk (horse/cow).

Coffee with milk is more concrete. In the second half of the 17th century the turkish ambassador introduced coffee to the court of Louis the XIVth, it became a smash hit and coffee houses became the next best thing. However french noblemen did not find the turkish brewing method palatable (bits of ground coffee sticking between the teeth) and set about to use filter bags and add cream milk and sugar (both outrageously expensive products at the time) to the beverage.On the other side of the continent the Ottoman Empire was besieging Vienna, and a rather resourceful person, a polish fellow named Franz Georg Kolschitzky offered his services to the city's defenders. He spent 10 years of his life in Istambul, spoke perfect Turkish and possessed a perfect moustache for the job. He donned turkish garments and snuck into the Ottoman camp, gathering vital information about the turkish plans. Supposedly based on his reports, the prince of Lorraine and the Austrians managed to attack and defeat the Ottomans. For his services, he received his share of the loot: 500 bags of green coffee beans, and permission to open up a coffee shop in Vienna. Since he became a local legend his shop had the right guests: nobles and the cream of the empire. However, the Austrians did not really like his turkish-style coffee, so opposed to his traditional methods other people seeing the demand for the drink increase, tried different methods to prepare coffee. As time went on more and more coffee shops opened in Vienna. From the 1700's you could have the "legendary" Wiener Mélange (espresso with honey and steamed milk), the Franziskaner (espresso with sugar and whipped cream) and the Kapuziner (a mélange where the steamed milk is mounded up to resemble the hoods of a certain order of monks).