Salt and pepper the chicken breasts and grill them in a frying pan with a dash of oil over medium heat until they are fully cooked. Chop them up and set them aside.
Wash the mushrooms and chop them up. Dice the onion and clove of garlic. In the same pan you used to grill the chicken breasts, sauté the diced onion and garlic with a generous dash of oil for about ten minutes over low heat. Add the mushrooms and white wine. Cook over medium heat until the alcohol evaporates. Away from the heat, mix the mushrooms with the cooked chicken, sliced ham and tomato sauce.
Make the béchamel. Melt the butter in a pan and add the flour. Cook it for about three minutes until it loses the flavor of raw flour, stirring continuously to avoid burning. Pour in the warm milk all at once. Add salt and pepper and whisk until it thickens, over medium heat.
Boil the sheets of cannelloni pasta according to package instructions. Cool and drain them and assemble the lasagna immediately.
In a dish suitable for the oven, add a tablespoon of béchamel to the bottom and a layer of cooked cannelloni sheets on top. Then add a layer of filling, another of cannelloni sheets and another layer of béchamel; then another of filling, another of cannelloni sheets and finish with béchamel and grated cheese.
Bake at 180 ºC for about 20 minutes to heat it up again and finish with a blast of the oven to brown the cheese.