The CPS Education Supplement provides some information related to the timing of degrees, such as EDYRDEG (year received most recent degree). Unfortunately, the answer choices for this question (and others similar to it) do not provide an exact year but rather reference the current year or some other previous year. You can see this question (YRDEG) and its answer choices from the 2017 Education Supplement Codebook here on page 8-4.
The full list of available variables through the education supplement can be found here.
Similarly, as you have noted, there is nothing in the ASEC or basic monthly samples that gathers the exact year of completion of an individuals highest educational attainment.
Since there was a change in the structure of obtaining information about education attainment in 1992, EDUC is designed to help bridge the gap and enable users to study educational attainment for the entire span of CPS data. While it may not be perfect, by combining HIGRADE and EDUC99 general comparisons can be made over time, and as much information as possible from both variables available in their respective time periods is retained.