Prepare the Dough: In a large mixing bowl, combine the flour, yeast, and salt. In a separate bowl, whisk together the warm milk, olive oil, and egg. Add the wet ingredients to the dry ingredients and mix until a shaggy dough forms.

Knead the Dough: Turn the dough out onto a lightly floured surface and knead for 5-7 minutes, or until smooth and elastic. Place the dough in a lightly oiled bowl, turning to coat. Cover with plastic wrap and let rise in a warm place for 1-1.5 hours, or until doubled in size.

Prepare the Chicken Alfredo Filling: While the dough is rising, prepare the filling. Heat olive oil in a large skillet over medium heat. Add minced garlic and cook until fragrant (about 30 seconds). Stir in heavy cream and cream cheese. Cook, stirring constantly, until the cream cheese is melted and the sauce is smooth. Remove from heat and stir in Parmesan cheese, parsley, salt, and pepper. Add the shredded chicken and stir to combine.

Assemble the Monkey Bread: Punch down the risen dough and divide it into small, bite-sized pieces (about 1-inch each). Roll each piece into a ball. Dip each ball into the chicken Alfredo mixture, ensuring it's well coated.

Layer in the Pan: Place the coated dough balls in a greased 9-inch bundt pan or tube pan, layering them evenly.

Second Rise: Cover the pan with plastic wrap and let rise for another 30 minutes.

Bake: Preheat oven to 350°F (175°C). Bake for 35-40 minutes, or until golden brown and cooked through. If the top starts to brown too quickly, cover loosely with foil.

Cool and Serve: Let the monkey bread cool in the pan for 10 minutes before inverting it onto a serving plate. Serve warm.