Are there any Extraordinary Claims for which Extraordinary Evidence has been discovered?

Yes! In fact, a great many of the claims now accepted by science would once have been thought extraordinary, but they are accepted because there is an extraordinary amount of evidence to support them. The transmission of sound and video via radio waves, for example, would have seemed at least as extraordinary as the claims of clairvoyance prior to the groundbreaking experiments of Hertz, Marconi, Baird and other scientists and inventors. Although we now take these inventions for granted, the first telegraph, radio, and television transmissions were regarded as quite astonishing in their time. This extraordinary evidence rapidly persuaded the world of the reality of wireless communication.

Other notable extraordinary claims for which extraordinary evidence exists include:

• Relativity Theory. There can be nothing more extraordinary than the claims made by Einstein’s special and general theories of relativity. “Time and distance are relative to the observer”; “gravity warps space and time”; “E=mc2“- these are claims that few of Einstein’s predecessors could have countenanced. Yet the evidence has matched the claims in its extraordinariness: from the observations of light bending as it passed the sun that first confirmed the general theory of relativity, to the horrific demonstration of Einstein’s famous equation at Hiroshima and Nagasaki, to experiments showing atomic clocks running at different rates at different altitudes (important for GPS), countless experiments have produced  results inconceivable without relativity theory.
• Evolution. Creationists are right about one thing (this may be all they are right about): the theory of evolution is an extraordinary claim, and should not be believed without extraordinary evidence to support it. The idea that the incredibly complex and almost perfectly functioning systems found in all living beings could have developed through the blind operation of natural processes is difficult to believe – which is one reason it took someone with the brilliance of Darwin to come up with the idea. But the evidence which Darwin was able to draw together points emphatically in favour of his theory, and the huge mass of evidence gathered since, from fossils to DNA and beyond, shows overwhelmingly that the theory of evolution is correct. The evidence here is extraordinary in (at least) two ways: it is extraordinary in the sheer amount of consistent evidence from a broad array of sources and discipline; and it is extraordinary in how unlikely such evidence would be if the theory of evolution were false.
• Plate tectonics. Here is a theory so extraordinary that, in the absence of extraordinary evidence, was largely rejected for half a century after it was proposed. When Alfred Wegener proposed the theory of continental drift in 1912 (others had suggested the idea as early as 1596), he had little evidence to support his hypothesis beyond a remarkable similarity between the coastlines of Africa and South America, and some suggestive fossil evidence. But without much better evidence, the theory was regarded as outlandish by the majority of Wegener’s peers, because it simply seemed impossible for something as large as a continent to move around, absent a plausible mechanism. But by the 1960s, strong new evidence of drift emerged through the study of mid-ocean ridges and magnetic striping, along with plausible mechanisms for crustal movements, and the scientific community came to accept the new theory of plate tectonics. Moreover, the evidence for plate tectonics has continued to grow as scientists learn to probe deeper into the Earth’s crust.
• Global Climate Change. The idea that humans could, by their collective actions over many years, fundamentally alter the climate of the planet is perhaps the most extraordinary claim of our age. After all, for most of human history, only gods were thought capable of controlling the weather. it is for this reason that the current consensus among  nearly all of the world’s climatologists that mean global temperatures are steadily rising as a result of human activity was a consensus that took a long time to build. The evidence, in this case, is extraordinary in its breadth and in its detail. This evidence comes in the form of terrestrial temperature records from all over the globe, satellite measurements of temperatures in the lower troposphere, and other records such as receding sea ice, showing a significant recent increase in average global temperatures. To this evidence is added a large body of evidence, using multiple methodologies, used to reconstruct a temperature record going back millenia, and sophisticated models used to project future tends. All of these sources of evidence combined indicate that the recent warming pattern represents an extraordinary departure from historical trends, which would be inexplicable if the hypothesis of anthropogenic global warming were not correct.