I recently heard about the deep watering method working for tomatoes, and was wondering if it also works for things like egg plant, and jalapenos.
A way to think about this question is to consider where the water will be compared to where the roots are. In order for the plant to benefit from watering, it must have roots at the same depth as the water. A large tree does typically have some deep roots. A seedling doesn't.
Because deeper soil takes longer to dry out, deep roots provide a plant with some protection against dry periods. That's a good reason to want roots to grow deeply. The plants will do better even on a hot dry day, you don't have to do your watering as often, and less moisture evaporates from the surface. There are other techniques with similar logic, like mulching. Moisture will move upward from depth within the soil profile, but much more slowly than water can move downward with gravity's help.
Once a plant becomes well established, its roots exist within a range of depths. Most plants have plentiful shallow roots, with root density declining fairly quickly with depth. Specifics depend on lots of factors - soil texture, species, plant age & health, etc. During the growing season, plants are creating new fine roots much of the time. Those roots branch outward from existing roots, and preferentially at the depth where moisture is optimal - not into standing water, but almost any plant likes ample moisture. So if the soil moisture is best toward the bottom of their current rooting zone, they will tend to grow their new roots deeper. If the opposite is true, with upper layers of soil moist and the deep soil dry, they will tend to grow shallow roots, all else being equal.
Part of "all else being equal" is that each species has its own tendency to grow deep or only shallow roots. Annual food crops tend to have shallow roots. Trees and other perennials tend to have some deep roots.
So first consider where your plant already has roots. Then if you can get the soil moist down to the plant's lower roots, that's a good thing.