In a large pot, place the whole chicken and add enough water to cover it completely. Bring to a boil over medium-high heat, then reduce heat to medium-low. Simmer for 1 hour to 1 hour 30 minutes until fully cooked (internal temperature reaches 74°C / 165°F). Remove the chicken and set aside to cool. Once cooled, remove the meat from the bones, shred into pieces, and discard bones and skin. Strain and reserve the cooking liquid.
Drain the canned peas, asparagus, and mushrooms, reserving the liquid from all three cans.
In a large pot or saucepan, melt the butter over medium heat. Add the flour and stir constantly for 1-2 minutes to form a smooth paste (roux).
Slowly pour the reserved canned vegetable liquid into the roux while stirring continuously. Bring to a boil over medium-high heat, then reduce heat to medium.
Simmer for 2-3 minutes, stirring occasionally.
If you prefer a thinner sauce, add some of the reserved chicken cooking liquid a little at a time, stirring until you reach your desired consistency.
Remove from heat and stir in the processed cheese until completely melted.
Add the drained peas, asparagus, mushrooms, and shredded chicken meat to the sauce. Stir gently to combine.
Stir in roughly half of the parsley and chives, or add more to taste.
Season with a pinch of salt and adjust seasonings to taste.
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