In a large skillet, heat 2 tbsp. EVOO over medium heat. Add the onions and season with salt and pepper. Cook, stirring occasionally, until the onions are caramelized, 15 to 20 minutes. Stir in 1/4 cup water, increase the heat and boil until the liquid is almost evaporated.
Meanwhile, in a large saucepan, heat the remaining 2 tbsp. EVOO over medium heat. Add the peas and thyme and season with salt and pepper. Cook, stirring occasionally, until the peas are heated through, 8 minutes. Stir in the lemon juice.
Heat a grill pan over medium-high heat. In a medium bowl, combine the beef, garlic, 3/4 tsp. salt and 1/4 tsp. pepper. Form the mixture into 12 patties about 1/2 inch thick. Grill the patties, flipping once, for 3 to 4 minutes for medium.
Place a patty on each roll. Top with the caramelized onions, cheese and roll tops. Serve with the peas.