A verb for “The cartographer painstakingly ___ a map of the city.”

Solution 1:

"crafted", or "created". The word Cartographer implies that the person has a certain skillset and they were employing those special skills in creating the map. "Plotted" is also a good word depending on how much focus you want to put on the technicality of creating the map. "contrived" or "derived" might also bee good choices depending on the context of the sentence.

Solution 2:

Make or draw a map appear to be the more common expressions:

Cartography (from Greek χάρτης khartēs, "papyrus, sheet of paper, map"; and γράφειν graphein, "write") is the study and practice of making maps.

(Wikipedia)

Cartography:

the science or art of making or drawing maps.

(Cambridge Dictionary)

Canyon Cartography

How The Map Was Made - Canyon Cartography – Wrightwood, CA Chris Kasten Cartographer Drawing map.

Solution 3:

You could use the verb chart. The OED offers this definition:

a. trans. To make a chart of; to lay down in a chart; to map.

Example citations match closely to the usage you are seeking:

The idea of actually charting these profound regions.

• 1851 J. P. Nichol Archit. Heavens (ed. 9) 114

Usually the object of chart is the subject that is being mapped itself, e.g.

The cartographers carefully charted the city

However, using "chart" in your example sentence would sound perfectly natural and express the meaning that you're looking for.

The cartographer charts a map of the city.

Based on the comments to my answer, there seems to be some contention around the notion that chart a map sounds natural. Having already mentioned that typically the object is what is being mapped itself, I still maintain that chart a map is acceptable on the grounds that it is easily understood and can be found in use by reputable writers.

Consider this passage from PBS News Hour, written by Nsikan Akpan:

The team examined median income data for individual areas in the U.S. to chart a map of the communities most at-risk for water poverty.