How to Make It
Add the cherry tomatoes halves and sliced jalapeños to a medium bowl and set aside.
Place the butter in a small heavy-bottomed saucepan over medium heat, stirring frequently, about 5 minutes. When the foam subsides and the color turns to a golden brown, turn off the heat. Using a spatula, quickly scrape and pour the browned butter over the bowl of tomatoes. Stir in the lime juice and chopped cilantro to combine. Season with salt and set aside.
In a non-stick skillet, heat 1/2 teaspoon of the oil. Place one tortilla in the skillet and scatter 1/4 cup of each of the cheeses evenly over the top. Top it with the second tortilla and cook over moderately high heat until crisp on the bottom, about 3 minutes. Flip it over and cook for another 2-3 minutes until the cheese is melted and the second side is golden. Repeat with the remaining oil and second quesadilla. Slice the quesadillas into wedges and transfer to plates. Spoon the salsa over the top and serve right away.