This sounds like a case where you are using an unregulated power supply, and it is expected to provide 7.5V only at its rated 1000 mA. The Arduino Uno and Ethernet shield use less than 1000mA, so the power supply, being unregulated, is supplying too high a voltage.
An Arduino Uno has a 5V regulator near the DC power plug, and it should be capable of handling 6 to 20 V input.
The voltage regulator will get warm when it's converting excess power to heat, and indeed it may seem excessively hot. Given that the voltage you measured is 10.6, there must be a lot of current flowing in order for it to get so warm. (Otherwise it would remain cool.)
The excessive heat seems to indicate a problem such as a short or failed component. Does this problem happen with a different power supply? Have you measured the current use with the power supply you are using now?
- Power the Arduino and shield from a USB supply, it's regulated 5 volts and bypasses the Arduino's on-board voltage regulator.
- Use a different power supply, preferably one that's regulated to the Arduino's input voltage range (recommended 6-12 on the product page).
- To really know what's going on, you need to measure current, and I would also check the input and output voltage of the on-board regulator. However, continuing to use the power supply that causes excessive heat may permanently damage your device(s).