Rinse the anchovies, pat them dry, and place them on a plate. Season them with salt and pepper, sprinkle on the thyme leaves, and drizzle with 1/4 cup olive oil. Set them aside to marinate while you make the salsa.
Combine the lemon supremes, red onion, chile, mint, and cilantro in a bowl. Add a pinch of sugar and some salt and pepper. Stir in 1/4 cup olive oil and mix well.
Heat the grill or a grill pan. Grill the anchovies, carefully turning them over halfway through cooking, for about 10 to 12 minutes, or until cooked through. Remove to a platter and top with the lemon and mint Salsa.