Heat butter and oil together in a skillet over medium heat; add onion. Cook and stir until onion is translucent, 10 to 15 minutes. Stir garlic, thyme, salt, and pepper into onion; cook and stir until fragrant, about 2 minutes. Remove skillet from heat.
Roll out puff pastry on a lightly floured surface. Line a baking sheet with parchment paper and place puff pastry on parchment paper. Place baking sheet in the freezer.
Preheat oven to 450 degrees F (230 degrees C).
Arrange tomato slices on a wire rack placed on top of a baking sheet. Sprinkle salt over tomato slices and set aside to drain tomatoes, about 10 minutes.
Mix Parmesan cheese, bread crumbs, feta cheese, and parsley together in a bowl. Spread 1/2 of this cheese mixture over the puff pastry, leaving a 1-inch border around the edges. Layer the onion mixture over cheese mixture layer.
Blot tomatoes dry and arrange over onion layer. Sprinkle remaining cheese mixture over tomato layer. Fold the puff pastry border over the edges of the tomatoes and lightly brush egg over pastry.
Bake in the preheated oven until crust is golden and puffed, about 20 minutes. Sprinkle fresh basil over galette and allow to cool for 15 minutes before slicing.
