The answer to your question is "not really".
As you mentioned in the question, Maillard reaction begins with carbohydrates and aminoacids. Meat is rich in aminoacids, so you don't need to add it to your recipe, it is already there.
If you want to increase the Maillard reaction, you need to use a marinade that is rich in carbohydrates, the simpler the ...
Two things that encourage Maillard reactions are more heat and higher PH. Modernist cuisine has instructions to caramelise food in water using a pressure cooker and baking soda. Baking soda can taste soapy. You can also look to non-flavored antacid from the chemist and something used in asian noodle making called alkaline salts or lye water.