According to the review of Valenta et al., nuts and peanuts give rise to IgE-mediated allergy, which is indeed a type of reaction that is subject to a form of sensitization. Although the process mostly occurs early in life, it seems that some level of "boosting" can happen on repeated exposures:
IgE-associated food allergies appear to develop early in childhood. This process is termed allergic sensitization. (A) Allergen contact via the gastrointestinal tract, via the respiratory tract, and eventually via the skin induces IgE production (primary sensitization) in genetically predisposed individuals. Repeated allergen contact activates allergen-specific T cells and induces IgE responses during the secondary immune response. [...]
In atopic individuals who have a predisposition toward developing IgE-associated allergies, encounters with allergen activate, after processing by antigen-presenting cells (eg, dendritic cells or B cells), allergen-specific T-helper 2 (Th2) cells, which produce cytokines such as interleukin (IL)4 and IL13. These cytokines induce class switching and production of allergen-specific IgE. Primary allergic sensitization (such as a class switch toward IgE production) occurs early in life and leads to T-cell and IgE memory, which can be boosted with repeated allergen contact (secondary immune response). Upon contact with a primary food allergen, nonallergic individuals produce allergen-specific IgG and IgA, which do not induce allergic reactions. The formation of food allergen–specific IgE is a main feature of IgE-associated food allergy and its diagnosis. [...]
Milk, eggs, wheat, peanuts, nuts, sesame, fish, fruits, and vegetables are common inducers of IgE-associated food allergy.
This is alas mostly a qualitative explanation (mechanism analysis). Unfortunately all of the (quantitative) boosting studies cited in that review were to respiratory allergens [quite a number of these are IgE-mediated as well].