Peel the onion and garlic, wash and finely chop. Grate the cheese. Wash the dill and chop finely. Preheat the oven to 180 degrees. Prepare a rectangular pan with high sides.
Prepare the meat filling. Heat vegetable oil in a deep frying pan, add onion and fry over medium heat until browned. Add minced meat, salt, pepper to the onion and mix. Continue frying over medium heat. In a separate container, mix tomato paste with water and pour into the minced meat. Add finely chopped garlic, Provençal herbs and stir. Cover the pan with a lid and simmer until half the liquid has evaporated.
Make bechamel sauce. Place a frying pan over medium heat, melt the butter in it, then add the flour and stir. Cook for 2-3 minutes, stirring constantly with a whisk. Add milk in a thin stream, constantly stirring the contents of the pan. Make sure that no lumps form. Bring the sauce to a boil, reduce heat and simmer until the sauce thickens. Add salt, nutmeg, stir and remove from heat.
Prepare lasagna. Grease the bottom of the pan with bechamel sauce and place the sheets of dough. Place a small amount of meat filling on top, sprinkle with grated cheese and cover with lasagne sheets. Repeat all steps. The final layers are bechamel sauce and grated cheese. Place the lasagna pan in a preheated oven for 30 minutes.
Before serving, sprinkle lasagne portions with chopped dill.