The flavor you want is from the high heat of the clay oven (around 900F, much like a wood-fired pizza oven). Probably nothing you can do to truly replicate it at home, but here are some things to try:
- Use a pizza stone in the oven, and turn the oven to MAX, as hot as it will go (above 500F). Let it preheat for 45 min at least to get the stone to full heat. Put the dough right on the stone (either by hand - risky but traditional, or with a pizza peel or upside down sheet pan covered in corn meal). The naan will cook much quicker, maybe 3-5 minutes at most. If you can get the oven hot enough, you'll get a little of the "char" that is part of the flavor.
- After mixing the ingredients, let the dough rest overnight in the fridge. The next day, pull it out, let it warm to room temp and resume the proofing process. That will let more sugars be created from enzymes converting starch to sugar, and will slightly improve browning (part of the flavor you are looking for).
Of course, if you want to get fun, you could look into one of the many tutorials on building a wood-fired oven in your yard. Lots of great baking to be had there, including a more traditional naan bread. For more tips, read up on all the things people do to their home ovens to bake better pizza - they will all apply to clay oven baking as well. Things like hearth inserts, faking out the "cleaning cycle" and other tips are common.
Based on comments and some more thinking, my first recommendation would be a grill - charcoal if you have it, gas if you don't. The procedure would be similar to making grilled pizza (well described in Peter Reinhart's "American Pie"). Using a charcoal grill with hardwood charcoal, a dough that is stiff enough to not immediately slide through the grate on your grill (may mean slightly less water in your dough), and making sure to brush the surface of the dough with oil or ghee before flopping down on the grill, you could probably get even closer to the clay oven. It still isn't the same thing, but you might get more of the smoky charred flavors you seek.