Heat a large skillet over medium heat and add the olive oil. When the oil begins to smoke lightly, shut off the heat and add the thyme leaves. The thyme should make a crackling sound as it hits the oil, cooking it slightly and mellowing the flavor. Add the bread in a single layer. Turn the heat back on and cook over medium heat until the first side browns, 2 to 3 minutes.
Use a pair of tongs to turn the toasts on their second side and season with kosher salt. Top each round with a slice of cheese and continue cooking them over very low heat or off the heat, an additional 2 to 3 minutes. Adjust the seasonings with sea salt, to taste. Serve alone or alongside soup, if desired.