I need to count how many times Dr. Abbott did a knee replacement, how many times Dr. Costello did a knee replacement, etc. Each query will reside in its own cell, so I need a formula for each query. In the above example, I would have four different cells containing the COUNT function/formula, e.g., one cell will be labeled "Abbott - Knee replacements", another will be "Costello -- Knee replacements", another will be "Abbott -- shoulder replacements", another will be "Costello -- shoulder replacements".
You should concatenate the two columns with doctor name and operation first in column C. To split out the number from the doctor name, you might want to use a string function like =RIGHT(cell, LEN(cell)-2). If the numbers go over 1 character in length you will need to replace the LEN part with a FIND function for the space character after the number.
After that you can used the COUNTIF function on the range of concatenations to return the number of unique doctor/operation pairs.