Cartographer
You can trade for 12 glass from a librarian for 3 emeralds, and then craft 32 glass panes with them, and you trade them with a cartographer with maxed out discounts, which gives you 13 free emeralds.
Cleric
You can begin the same process with the Cartographer, but instead of crafting glass panes, you craft 12 glass bottles, and then trade with a cleric with max level discounts, which gives you 9 free emeralds.
Farmer
Given that crops grow fastest when planted in rows, growing only a single crop is inefficient. So finding the top 2 cash crops is key.
While Beetroot and Wheat have a lower item/Emerald need, the crop yield is insufficient on a per farming tile ratio. Each tile farmed will yield only one food item and a multiple of seeds. While both Carrots and Potatoes will yield 3.43 crops per farm tile on average. Meaning that even with one Carrot/Potato being consumed for reseeding, an average of 2.43 crops are generated per tile. This does not even factor in harvesting using a Fortune enchanted harvesting tool, which will further increase yield with Carrot/Potatoes but not increase yield of crops (only seeds) with Wheat/Beetroot.
Of note: The Poisonous Potato is a bonus item, it is not generated in place of a regular potato
Bottom line
| Crop | Crops/Tile | Emeralds/Tile | Tiles/Emerald |
|---|---|---|---|
| Carrots | 2.428 | 0.11 | 9.05 |
| Potatoes | 2.428 | 0.09 | 10.70 |
| Wheat | 1 | 0.05 | 20 |
| Beetroot | 1 | 0.07 | 15 |
Librarian
Bookshelves can be sold by librarians, which can be destroyed to obtain 3 bookshelves, which then can be used to trade back emeralds if the librarian villager has maxed out discounts, giving you 2 free emeralds.