The problem with monochromatic painting is that white paint doesn't create half tones. My art teacher never tired to tell us that "white isn't a color, it's the absence of color." It took me longer than I'd like to admit to understand what she meant with that.
Looking at a classical color wheel we see the fully saturated colors at the edge and the same colors mixed with increasing amounts of white towards the center.
If you look at Ultramarine blue and try to lighten the color, the next lighter tone seems to be a mixture of Emerald green + blue instead of a mixture of Ultramarine + white. The same happens from Scarlet red all the way up to Cadmium yellow. The next lighter tone is always a mix with a different base tone instead of a mix with white.
Starting again at Ultramarine and moving towards the center of the wheel there is an increasing amount of white (or rather water in case of water colors) mixed into the paint. But instead of getting lighter, the color gets less saturated. It seems chalkier, maybe foggy or more pastel like, but not much lighter. The white paint decreases the color value instead of increasing the lightness. That's why "white is the absence of color".
I must admit that the color wheel above was most probably created using water colors and those behave different from oil paints. The closest thing created with oil paints I could find was this mixing chart by Natural Earth Paint Canada (Click the link to see a bigger version of the image in their web shop. I'm not affiliated with them in any way.)
The image quality is very bad, but the color tones should be clear enough. Each tone is given in its fully saturated form and mixed with Titanium white below that. There are different white pigments in paints, the most common being Titanium white and Zink white. The same base tone mixed with Zink white would look different from the one mixed with Titanium white. There are even mixes of Titanium and Zink white available that create a slightly different effect.
And let's not forget that the quality of your paint can also play a huge role.
My advice is to create a similar mixing chart with the colors you have. Mix one base tone (for example Burnt umber) with all variations of black and white you have. Then mix the same base tone with the 3 primary colors. Create a small color field or a gradient (white - color - black) on a sturdy piece of white cardboard to keep it as a reference. Just experiment and see how your paints react to each other.
I found a good example of monochromatic portrait painting by Zin Lim on Youtube. You'll see that they use Cadmium yellow in addition to Burnt umber, Ivory black and Titanium white. The pure Burnt umber is actually the dark tone. To create lighter tones, they mix it with cadmium yellow and a hint of white. To create highlights they mix more white and only a hint of yellow into the umber. That makes the highlights lose color value (they get chalky or muddy), but in proximity to the saturated tones that were lightened with yellow it looks natural.
In my personal opinion, that's the best way to create monochromatic paintings. The result should matter more than the materials that went into it. I could use 10 different paints of a red tone and still call the result a "monochromatic painting".