Make the pesto by blending the basil, Parmesan, pine nuts, garlic, pepper, and 1/4 cup olive oil in a mini food processor (or do it by hand with a mortar and pestle, if you're old-school) until creamy. Add kosher salt to taste.

Line two of the bread slices with mozzarella and spread pesto on the other two bread slices, then put them together to make two sandwiches.

Whisk the egg and milk together in a shallow bowl that's wide enough to fit the whole sandwich in. Reserve.

Heat the butter and remaining tablespoon olive oil in a large cast iron or nonstick skillet over medium-low heat until the butter is melted and foaming.

Dip both outer sides of a sandwich in the egg-milk mixture, then place in the pan. Repeat with the remaining sandwich.

Cook, flipping once or twice, until the mozzarella melts and the bread is golden and toasty. Serve immediately.