In a small bowl, stir together 1/2 tablespoon melted butter and 1/2 tablespoon olive oil. Add the pinch of salt and mix.
Brush one side of each of the 2 slices of sourdough bread with the butter–oil mixture.
Heat a nonstick skillet over medium heat.
Place both sourdough slices in the skillet, buttered side down.
While the bread is warming in the skillet, on one of the slices (the one you will use as the bottom), layer the 2 slices of pepper jack cheese with very little overlap so the cheese mostly sits on the bread.
Arrange the 4–5 slices of ripe tomato in a single layer on top of the pepper jack.
Place the 1 3/4 ounces of feta (cut into thin slices) evenly over the tomato slices.
Sprinkle the 1/2 teaspoon za'atar spice blend over the feta.
Use a spatula to lift the other warmed slice of sourdough from the skillet and place it on top of the filling, with its buttered and browned side facing upward.
Press down gently with the spatula and grill until the bottom of the sandwich is nicely browned and golden, then carefully flip the sandwich and grill the other side until it is browned and the cheese is melted to your liking.
Remove from the skillet and serve warm. (Alternatively, assemble and grill the sandwich in a panini press.)