I accidentally mixed a dry spoon that was used to mix chicken soup into a pot of veggies being cooked on the stove. I ate some of the veggies. Am I now fleshig?

Shalom U'vracha, You are not fleishig. Explanation: If the spoon was clean it is considered 'noten taam bar noten taam leehetera'--meaning that we are only talking about an indirect flavor. The meat gave flavor to the spoon, which gave flavor to the soup. Even if the spoon had a bit of leftover from the soup it is 'batel b'shishim', which means it is insignificant (literally cancelled in one sixtieth). L'catchila, (or done optimally), if you want the soup to be parve, then don't mix a fleshig spoon--just to play it safe. All the best