Heat a grill pan over medium-high heat.
Drizzle the melon slices with the olive oil and place in the grill pan. Grill undisturbed until browned on the first side, about 30 seconds. Flip and grill on the reverse side for an additional 30 seconds. Remove to a cutting board to cool slightly, then cut into 1/2-inch pieces. Place the grilled melon in a medium bowl, along with the tomatoes, mint, lime juice, chile paste and 1/4 teaspoon salt; mix well with a spoon. Set aside to let the flavors mingle.
Meanwhile, heat 5 tablespoons water in a large skillet over medium heat. When bubbles just start rising to the surface, begin whisking in the butter, 1 tablespoon at a time, whisking constantly to form an emulsion. When all of the butter is incorporated, reduce the heat to medium low to keep the mixture at just below a simmer. Add the thyme and 1/2 teaspoon salt, swirling the pan to combine. Add the asparagus and toss gently to coat. Cover the pan and cook until the asparagus is tender all the way through when pierced with the tip of a knife, 5 minutes.
Remove the asparagus to a platter and spoon some of the melon salad over the top. Top with a few shavings of ricotta salata if desired. Serve with additional salad on the side.