Heat a 12-inch cast iron skillet over medium-high heat. Add prosciutto to the skillet and cook until crispy, stirring occasionally, about 2 minutes.
Remove prosciutto from skillet and set aside.
Add 1/4 cup butter to the same skillet. Let over medium heat and continue to cook for about 2 to 3 minutes, stirring constantly.
Once the butter is brown and smells nutty, remove from heat to a small bowl.
Add remaining butter and melt. Add gnocchi and cook over medium heat in the skillet until browned, about 5 minutes.
Stir in garlic and thyme leaves, cooking until fragrant, about 1 minute.
Stir in salt, pepper, peas, and asparagus, along with half of the brown butter. Cook until vegetables are bright green and cooked through. This will take about 2 to 3 minutes.
Stir in remaining brown butter and lemon juice, cooking until fragrant, about 1 minute.
Stir in 1/4 cup parmesan cheese and prosciutto.
Serve immediately with remaining parmesan cheese.