Medieval society determined the date of Easter (or Passover) relative to the Full (or New) Moon on or after the (day of the) Vernal Equinox, where the state of the Moon was observed rather than calculated.
Allowing that this was an era in which the remainder of the calendar was somewhat fuzzy, how was the precise day of the Vernal Equinox determined?
People have been able to determine the approximate date of the equinox since ancient times. The Babylonians even knew (to fairly good precision) the location of the First Point of Aries (the March equinox point) on the celestial sphere, where the ecliptic plane crosses the equatorial plane. Hipparchus was able to determine that the equinox point had precessed slightly in the intervening centuries.
Alexandrian mathematicians were aware that the daily path of the shadow cast onto a plane by a point is (approximately) a hyperbola. It's not exactly a hyperbola because the Sun's declination changes slightly over the course of the day. The great geometer Apollonius studied the hyperbola, and other conic sections.
Sundials often include such declination lines, which can be used to determine the date, and / or when the Sun enters the various signs of the Zodiac. Even quite ancient dials feature declination lines.
Here's an example of declination lines on a vertical sundial plate from Carl Sabanski's The Sundial Primer, which has a wealth of information about sundials.
The declination lines are the coloured curves. (The figure-8 shape is a noon analemma, which can be used to convert sundial time to mean time).
At the equinoxes, when the Sun's declination is zero, the declination curve degenerates to a straight line.
Of course, the declination curve looks fairly straight on the days near the equinox. However, the rate of change of the declination is at its fastest near the equinox, so you can make a pretty good estimate of the equinox day by making shadow point plots for a week or two and interpolating.
Also, on the equinox, the sunrise and sunset occur at 6 o'clock (solar time).
Determining the time of the equinox to higher precision requires more sophisticated techniques. There's some information about that at How was the First Point of Aries measured in ancient times?
The precise time of the equinox is tricky to calculate because the Earth's motion is affected by the Moon.
As noted in comments, Easter has been calculated, not measured since the third century. However measuring the date of the equinox is something that is well within the abilities of any ancient astronomer.
You need a fixed spot to observe from, with a clear easterly horizon. You then note the direction of sunrise over the course of a year. The direction of sunrise moves by about a degree each day in March, or about two sun-diameters. You then note when the sunrise is due East. Or if you are "fuzzy" on directions, then the date halfway between the direction of the solstices. These kind of measurements can be done without sophisticated tools. Even if you are plagued by clouds, you would be able to predict (with good confidence) the date of the equinox from many weeks in advance.
Even if you don't have a completely clear horizon, if you can find the date of the vernal equinox from someone who does know, you can then find the direction of sunrise on that date. This could be as easy as "from the top of the church tower, on the vernal equinox the sun rises in the direction of yonder tall pine tree." This is now a simple observable (until yonder pine tree falls down!)
