The lacy –leafed, tap-rooted plants grew first on the windswept highlands of Afghanistan.

The Greeks used this plant as a digestive tonic when they had overeaten other foods.

The plant being described here is most likely a type of a vegetable or herb that has lacy leaves and a taproot. To determine the specific name of this plant, we can follow a step-by-step analysis:

1. Identify the characteristics of the plant:
- Lacy-leaves: This suggests that the leaves have intricate or delicate structures, possibly resembling lace.
- Tap-rooted: Taproots are long, thick primary roots that grow vertically into the ground.
- Originates in the windswept highlands of Afghanistan: This tells us the plant's place of origin.

2. Search for plants with similar characteristics:
Using a search engine or a botanic database, we can search for plants with lacy leaves and a taproot, paying particular attention to plants native to Afghanistan.
Here are a few examples of plants that fit the description:
- Parsley: Parsley has lacy, fern-like leaves and a taproot.
- Dill: Dill also has delicate, fine leaves and a strong taproot.
- Fennel: Fennel has feathery, lacy leaves and a taproot.

3. Analyze the historical context:
The clue about the Greeks using the plant as a digestive tonic provides additional information. To narrow down our search, we can focus on plants that were commonly used by the Greeks for digestive purposes.

4. Determine the likely plant:
Considering the historical use and the characteristics of the plant, fennel seems to be the most suitable option. Fennel has a long history of use as a digestive aid and its feathery, lacy leaves match the description given.

Therefore, based on the information provided, the plant in question is likely "fennel."