Cook Watermelon: Cut watermelon into 1 inch (2.5 cm) thick slices and remove the rind. Brush each piece with 1 Tbsp olive oil. In a grill pan, saute pan, or on the grill, cook watermelon at medium/high heat until char lines develop and watermelon lightly browns, about 3 minutes each side. Remove from heat and chop into large chunks.
Assemble: In large bowl, gently combine watermelon, feta, arugula, red onion, mint, and basil. Whisk together 1 Tbsp olive oil, balsamic (more to taste), salt, and pepper. Gently combine with the rest of the salad. Serve warm or chilled.